There are at least two ideas involved.
First is that the expansion of the universe is not linear. While the Big Bang happened around 14B years ago, that does not mean that 13B years ago, the Universe is 1/14th of its present size. Current theory suggests that a large portion of the cosmological inflation (where the Universe increased by 26 or more orders of magnitude in linear dimensions) happened within much, much less than a second after the Big Bang. And as another example, the current theory estimates that at the time that the cosmic microwave background was emitted (which was about 0.5 million years after the birth of the Universe, placing it about 1/30000 the current age of the Universe), the universe is already about 1/1000 its current size (in length).
Second is that the apparent recession of far away objects from us is not so much objects flying apart from each other. Rather, it is space being added in between objects. Imagine you being the photon, and two turtles (moving slower than you) being the galaxies. Put turtle one in the first carriage of a train, and put turtle two on the 10th carriage of a train. And you start walking. Say it takes you 1 minute to traverse a carriage, and it take the turtles 10 minutes. Then in the case where the turtles walk away from each other, it will take you a bit under 12 mintues to get from the first turtle to the second (you walk 10 minutes to the tenth train, and the turtle has gotten to the 11th. You walk another minute to the 11th train. The turtle is just a few steps in front of you.)
But that's not how the universe expands. The expansion of the universe is more like the following: suppose every 6 minutes, all the carriages decouple, and between each pair of the original carriages plops one more car! So you walk for 6 minutes (having traversed 6 cars), and you look up, and see that the second turtle is 8 cars in front of you (and the first turtle is 12 cars behind). And you walke another 6 minutes. Plop comes the extra cars, and now you are 4 cars from the second turtle and 36 cars from the turtle behind. And finally after another 4 mintues you catch up to the second turtle.
From the point of view of the second turtle though, you would have travelled from a turtle that is now 40 cars away from him, while taking only 16 minutes! This ties back into the funny idea that light emitted from an object 13B lightyear away can take quite a bit less than 13B years to get here, due to the inflationary Universe.
This is why cosmologists and astronomers use red-shift to measure distance, because there is no reasonable intrinsic notion of distance that is free from ambiguity: should distance be described by how far away the turtles are when you started walking? or when you finished walking? or the number of carriages you (the photon) have traversed? Instead of that, they measure it using red-shifts, which can roughly fit into this turtle-you framework as how flushed your cheek is from all that walking when you reached turtle number two. Based on the redness of your cheeks, the turtles can calculate how much you exerted yourself, and thus for how long you've been traveling, and using known rules of the addition of new cars (the value of Hubble constant), the turtles can estimate the distances to other turtles. :-)
(I'm going to skip discussion of standard turtles, which are turtles from which you will always depart well rested and not flushed, nor how the turtle simiano-ferroequinologists found out about their rates of locomotive expansion.)
Hubble's Law, when written in this form,
$$
v = H_0D,
$$
means: if $D$ is the current distance of a galaxy, and $H_0$ the Hubble constant, then $v$ is the current recession velocity of the galaxy. So it tells you what the recession velocity of a galaxy is right now, not what it was in the past.
Basically, the Hubble Law is a consequence of the cosmological principle, i.e. that the universe on large scales is isotropic and homogeneous. This means that the expansion of the universe can be described by a single function of time, the so-called scale factor $a(t)$, so that the distance to a faraway galaxy increases over time as
$$
D(t) = a(t)D_c,
$$
where $D_c$ is a constant, called the co-moving distance to the galaxy; $D(t)$ is known as its proper distance. Also, the present-day value of $a(t)$ is set to 1 by convention, i.e. $a(t_0)=1$, so that $D(t_0) = D_c$.
If we take the derivative, then
$$
v(t) = \dot{D}(t) = \dot{a}(t)D_c = \frac{\dot{a}(t)}{a(t)}D(t) = H(t)D(t),
$$
with $v(t)$ called the recession velocity and $H(t)=\dot{a}/a$ the Hubble parameter. This is the general version of Hubble's Law at cosmological time $t$, which at the present day takes the familiar form
$$
v = H_0D,
$$
where $v=v(t_0)$, $H_0=H(t_0)$ and $D=D(t_0)$. But in this form, Hubble's Law isn't very useful: it's a purely theoretical relation, because the recession velocity of a galaxy cannot be directly observed, nor does it say anything about the expansion of the universe in the past. It only tells us how fast a galaxy is moving from us right now, if you know its current distance.
However, there's a related quantity that we can observe, namely the redshift $z$ of a galaxy, which is the change in wavelength of its photons as they travel through the expanding space:
$$
1 + z = \frac{\lambda_\text{ob}}{\lambda_\text{em}},
$$
with $\lambda_\text{em}$, $\lambda_\text{ob}$ the emitted and observed wavelength respectively.
Unlike the recession velocity, the redshift does give us information about the past, because the redshift of a photon accumulates over time, during its journey from the source galaxy to us. By comparing the redshifts of two galaxies, we can deduce information about the expansion rate in the past: suppose we observe two galaxies with distances $D_1 > D_2$ and redshifts $z_1 > z_2$, which emitted their light at times $t_1$, $t_2$ respectively. Then the difference in redshift $z_1-z_2$ will tell you how much the universe expanded in the time interval $[t_1,t_2]$.
In other words, if the expansion of the universe were decelerating, we'd see that the redshift of distant galaxies accumulated a lot in the distant past, when the expansion rate of the universe was high. However, observations showed that the expansion of the universe first decelerated and then started to accelerate again (the transition from deceleration to acceleration occurred when the universe was about 7.7 billion years old). This means that there was a time when the expansion rate was at a minimum, during which the redshift of photons increased less.
The relation between $v$ and $z$ is determined by the cosmological model. In the Standard Model, it can be shown that the observational version of the Hubble Law looks like this:
$$
H_0D = c\int_0^z\frac{\text{d}z'}{\sqrt{\Omega_{R,0}(1+z')^4 + \Omega_{M,0}(1+z')^3 + \Omega_{K,0}(1+z')^2 + \Omega_{\Lambda,0}}},
$$
where $\Omega_{R,0}$, $\Omega_{M,0}$ and $\Omega_{\Lambda,0}$ are the fraction of radiation, matter and dark energy in the present-day universe, and $\Omega_{K,0} = 1 - \Omega_{R,0} - \Omega_{M,0} -\Omega_{\Lambda,0}$ describes the curvature of space.
Early observations and inflation models suggested that the curvature of space is close to zero, which would mean that $\Omega_{M,0}\approx 1$ if there's no dark energy (the contribution of radiation is negligible). On the other hand, dynamical studies of galaxy clusters indicated that $\Omega_{M,0}\approx 0.3$. Furthermore, models without dark energy led to a 'cosmic age' paradox: the calculated age of the universe in these models was less than the age of the oldest observed stars (see Krauss 1995 for a review). These issues were resolved in 1998 when two teams applied Hubble's Law to a sample of supernovae, comparing their distance and redshift, which offered clear evidence for dark energy, with $\Omega_{M,0}\approx 0.3$ and $\Omega_{\Lambda,0}\approx 0.7$, and a Hubble constant $H_0\approx 65\;\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$. These values have been further refined by CMB observations.
The effect of dark energy can be seen in this figure:
Here, I've set $\Omega_{R,0}=0$ and $H_0 = 63.7\;\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$ (the most recently obtained value). The red curve is a model with dark energy. As you can see, for a given distance the corresponding redshift is much lower than in models without dark energy, i.e. without acceleration.
Extra info
It's interesting to examine these models in more detail. Once the values of the cosmological parameters are fixed, the evolution of the universe can be calculated. In particular, the cosmic time can be written as a function of the scale factor:
$$
t(a) = \frac{1}{H_0}\int_0^a\frac{a'\text{d}a'}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a' + \Omega_{K,0}\,a'^2 + \Omega_{\Lambda,0}\,a'^4}},
$$
which can be inverted to yield $a(t)$, and thus also $\dot{a}(t)$ (see also this post and this post). The age of the universe is $t_0=t(1)$, and we find that $t_0 =$ 14.0, 11.8, and 9.7 billion years for $(\Omega_{M,0},\Omega_{\Lambda,0})= (0.3,\,0.7), (0.3,\,0.0), (1.0,\,0.0)$ respectively. In other words, dark energy increases the age of the universe (which also solves the age paradox previously mentioned). This is a crucial point, as I will show below.
Furthermore, there's a simple relation between the redshift of light and the scale factor: if a photon is emitted at a time $t_\text{em}$, then its redshift will accumulate as
$$
z(t) = \frac{a(t)}{a(t_\text{em})} - 1,
$$
so that its present-day observed redshift is $z = 1/a(t_\text{em})-1$ (see wikipedia for a derivation). In other words, the observed redshift of a photon tells us when it was emitted.
Let's apply this to a particular galaxy. Suppose we have a galaxy at a present-day distance $D = 10$ billion lightyears. We then have the following situation:
The first graph shows the proper distance of the galaxy and its light $D(t)=a(t)D$ as a function of lookback time $t_0-t$. In all three cases, $D(t)=0$ corresponds with the 'big bangs' of these models.
The change from dotted lines to solid lines indicate the moment $t_\text{em}$ at which the galaxies emitted the photons that we observe today; the dashed lines are the paths of those photons. In all three models, the photons were emitted about 7 billion years ago. But the corresponding scale factors $a(t_\text{em})$ are very different: $a(t_\text{em})=0.54,\,0.48,\,0.43$ for the red, blue, green models respectively. This is a direct consequence of the different age of the universe in the three cases.
This immediately explains the redshifts shown in the graph below: the present-day redshift of the light is $z=0.86,\,1.1,\,1.3$ in the respective models, i.e. the observed redshift is much lower in the dark energy model.
Although it's not very clear, the red curve of $D(t)$ has an inflection point about 6 billion years ago, corresponding with the moment when $\ddot{a}=0$ and the expansion of the 'red' universe began accelerating. This is much clearer in the top right graph, showing the recession velocity $v(t)=\dot{a}(t)D$. In all three cases, $v(t)$ was much higher in the past, which means that the expansion has been decelerating. But in the dark energy case, $v(t)$ reached a minimum value around 6 billion years ago, and began to increase again. This is the effect of recent acceleration due to dark energy.
However, note that $v(t)$ is much lower in the dark energy universe. Again, this is a consequence of the age of the universe in the models: it took 14 billion years for $a(t)$ to increase from 0 to 1 in the red model, while it took only 9.7 billion years in the green model. As a consequence, $\dot{a}$ is much lower in the former case.
Finally, the last graph shows the Hubble parameter $H(t)=\dot{a}/a$, showing that even in the accelerating universe $H(t)$ decreases.
To summarise, the influence of dark energy determines the redshift, proper distance and the recession velocity over time, but it's not really its effect on the accelerating expansion that's important, but rather its effect on the age of the universe.
As a final note, the proper distance of a galaxy isn't directly observed. It can be derived from its so-called luminosity distance (comparing the apparent brightness and intrinsic luminosities of supernovae). So we should actually compare the evolution of a galaxy with a fixed present-day luminosity distance in different models rather than a fixed proper distance, but this doesn't change the argument.
Best Answer
All of the universe which is observable, we can see :) But you're right---there's lots of additional universe out there that we can't see, and we'll never be able to. In fact, because of expansion, more and more of the visible universe is actually leaving the region which we can see---which is called our 'light-cone'.
Star with a point in space-time---called point 'A'. Now imagine light traveling away from that point in all directions. In the plot below, time is graphed on the vertical axis, and space is in the horizontal plane. Every second that goes by, the light goes 1-light-second (ls) further away. If you trace out the path of those light-rays, it defines two cones:
Everything in the cone behind point A is in its 'past', everything in the cone in front of it is in its 'future'. Everything outside of both cones is 'causally disconnected' (like point 'E'). Points outside of the cones will never interact with point 'A' (because information from them is limited to the speed of light).
If you imagine that you are at point 'A'. Then the visible universe is everything within your past-light-cone. The circle at the bottom would be the border ('event horizon') of your visible universe.
Because space-time is expanding, the shape of the 'cones' is actually bent, kind of like this:
But anyway, there is lots of universe outside of your light-cone (outside of out 'observable universe'), possibly infinite amounts of it (we don't know).