Consider this example: You get into your car and drive around. The function $f(t)$ tells you how much distance you have covered until time $t$. Now the derivative of this function is your momentaneous velocity.
So, since you only asked for an interpretation, think about this: Did you ever sit in a car wondering how fast you were going 'in that instance'? Of course, technically speaking, the tachometer only gives you some average speed over the last second or so but one still often thinks of it as the speed of the car 'right now'.
So, let's review the definition of a secant line, the definition of a tangent line, and the definition of the derivative:
A secant line to a curve is a line that intersects the curve in two distinct points.
The derivative of a function $f(x)$ at point $a$, denoted $f'(a),$ is defined to be the limit
$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}, $$
provided this limit exists.
A tangent line to a curve $f(x)$ at a point $a$, if it exists, is defined to be that line with slope $f'(a)$ going through the point $(a, f(a)).$
(Side note: a tangent line is most definitely NOT a line that intersects a curve at only one place. This is a common, but misguided definition of the tangent line often encountered in high school geometry textbooks and is incorrect in general, though true for conic sections such as circles, ellipses, hyperbolas, and parabolas. It's better to think of tangent lines in terms of the derivative.)
First off, notice that the derivative is defined as a limit. So, for the derivative to exist, the limit has to exist. There are multiple ways a limit in general can fail to exist (right-hand and left-hand limits not equal, infinities, etc.), and there are additional ways that a derivative-type limit can fail to exist. For example, while this is not sufficient for the derivative to exist, the function must be continuous at $a$. That means $\lim_{x\to a}f(x)=f(a),$ a statement which says three non-trivial things: 1. The limit exists. 2. $f(a)$ is defined. 3. The limit equals the value of the function. Any of those can fail. In addition, you could have, say, a corner at $a$. The function $f(x)=|x|$ has a corner at $a=0$. While the function is continuous there,
$$1=\lim_{x\to 0^{+}}\frac{|x|-|0|}{x-0}\not=\lim_{x\to 0^{-}}\frac{|x|-|0|}{x-0}=-1.$$
The function $f(x)=|x|$ is therefore not differentiable at $x=0,$ because the required limit does not exist. Geometrically, think about the corner this way: what unique value could you assign to the "slope" of $|x|$ at $x=0?$ You could draw any number of candidates. What the limit definition is saying is that you have to be able to come up with a unique value for the slope at a point, in order for the function to be differentiable at that point.
As for the relationship between secant lines and tangent lines, just notice that the slope of the secant line to $f$ at the points $x$ and $a$ is given by
$$m=\frac{f(x)-f(a)}{x-a}.$$
The slope of the tangent line to $f(x)$ at $x=a$ is defined as the limit of the secant line slopes. You can imagine, geometrically, that you're letting the point $x$ get arbitrarily close to $a$. In the limit, you get the tangent line (provided the limit exists).
Best Answer
Instantaneous rate of change of a differentiable function at a point is by definition the change in value of the function when the point is infinitesimally perturbed. The definition of instantaneous velocity at any point itself is the rate of change of position at that point, and is the velocity "at that point".
If $t$ denotes time and $f(t)$ denotes position at time $t$, then the velocity at time $t_0$ is defined as $$v_0 = f'(t_0)=\lim_{t\to t_0} \frac{f(t) - f(t_0)}{t-t_0}$$
As you mention, the limit is an "actual value", and may be defined based on $f$. When it is defined, it gives the exact velocity at time $t_0$. It is true that when you substitute $t_0$ in the limit, you get $\frac{0}{0}$, which is undefined. But the fact that a function is undefined at a point does not mean that the limit on approaching the point is undefined (take, for example, $\lim_\limits{x\to 2}\frac{x^2-4}{x-2}$). Moreover, substituting $t_0$ represents no change in time, whereas the velocity is defined for an infinitesimal change in time, for which the change in position is given by the limit, and is often well defined.
The key is that taking the limit allows you to exactly compute the change of the position for an infinitesimal change in time. Substituting $t_0$ represents no change in time, and substituting any non-infinitesimal change in $t_0$, say, $t_0+t_1$ gives the average velocity over the time period $t_1$.