The time dilation is due to a difference in the gravitational potential energy, so it is due to the difference in height. It doesn't matter whether the strength of the gravitational field varies, or how much it varies, all that matters is that the two observers comparing their clocks have a different gravitational potential energy.
To be more precise about this, when the gravitational fields are relatively weak (which basically means everywhere well away from a black hole) we can use an approximation to general relativity called the weak field limit. In this case the relative time dilation of two observers $A$ and $B$ is given by:
$$ \frac{dt_A}{dt_B} \approx \sqrt{ 1 + \frac{2\Delta\phi_{AB}}{c^2}} \approx 1 + \frac{\Delta\phi_{AB}}{c^2} \tag{1}$$
where $\Delta\phi_{AB}$ is the difference in the gravitational potential energy per unit mass between $A$ and $B$.
Suppose the distance in height between the two observers is $h$, then in a constant gravitatioinal field with acceleration $g$ we'd have:
$$ \Delta\phi_{AB} = gh $$
If this was on the Earth then taking into account the change in the gravitational potential energy with height we'd have:
$$ \Delta\phi_{AB} = \frac{GM}{r_A} - \frac{GM}{r_B} $$
where $r_A$ and $r_B$ are the distances of $A$ and $B$ from the centre of the earth and $M$ is the mass of the Earth. Either way when we substitute our value of $\Delta\phi_{AB}$ into equation (1) we're going to get a time dilation.
As for the accelerating rocket: the shortcut is to appeal to the equivalence principle. If acceleration is equivalent to a gravitational field then it must also cause a time dilation in the same way that a gravitational field does.
Alternatively we can do the calculation rigorously. The spacetime geometry of an accelerating frame is described by the Rindler metric, and we can use this to calculate the time dilation. The Rindler metric for an acceleration $g$ in the $x$ direction is:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 - dx^2 - dy^2 - dz^2 $$
We get the time dilation by setting $dx=dy=dz=0$ to give:
$$ c^2d\tau^2 = \left(1 + \frac{gx}{c^2}\right)^2c^2dt^2 $$
and on rearranging this gives:
$$ \frac{d\tau}{dt} = 1 + \frac{gx}{c^2} $$
which is just the equation (1) that we started with.
I am afraid your allegoric figuring out why gravity causes time dilation is further confusing. The physical reality is measured by the metric of a reference frame which composes the coordinates, but the single coordinates do not have necessarily a specific meaning in GR (general relativity).
The Einstein equivalence principle allows to describe gravity in terms of geometry (metric) of a curved spacetime.
If we consider a static (Schwarzschild) spherical mass, radially the curvature is different and the proper time of stationary observers, as given by the metric, is progressively slowing as you approach the mass, if compared to the time measured by an observer far away from the mass.
The time dilation of SR (special relativity) is still a different concept as it is symmetrical between two observers in uniform relative motion. Instead the gravitational time dilation is not symmetrical; in fact the stationary observer measures a time contraction if compared to the far away observer.
Best Answer
There is a common misconception that time dilation is some kind of active process i.e. something acts on the dilated observer to slow their time down. This is not the case. Instead time dilation exists because two observers measure time in different ways.
Assuming you are located on the surface of the Earth an observer on Pluto would observe your time to be dilated i.e. their clocks would run faster than yours. But does your time feel dilated to you? Your clocks still run at one second per second. Your radioactive nuclei still decay at the same rate. As far as you are concerned your time runs perfectly normally.
We measure time and space by using some set of coordinates. Typically we measure position in space by setting up $x$, $y$ and $z$ axes, and to measure time we use a fourth $t$ axis. This makes for a four dimensional graph, which is a bit hard to visualise but mathematically this is straightforward. Then we can track the flow of time by how fast objects move along the time axis. For more on this see: What is time, does it flow, and if so what defines its direction?
The observer on Pluto also uses coordinates to measure time, but due to the curvature of spacetime their time axis is not the same as your time axis, and this means they measure time differently. This is why the observer on Pluto observes your time to be dilated.
So the answer to your question is that time dilation does not affect matter in the sense of some active mechanism acting upon the matter. All it means is that different observers measure time differently because their definitions for the time axis are different.