How does gravitational time dilation affect matter? Einstein successfully melded together space and time into one entity called spacetime in his General relativity theory and gave us further insight into how matter affects spacetime.
John Wheeler said “Spacetime tells matter how to move and matter tells spacetime how to curve” but does this simple description of Einstein’s theory tell us what spacetime is doing to the internal mechanisms of matter. How it is affecting the internal clock that exists within matter. Are the oscillating atoms actually slowed down by the effect of time dilation resulting from a greater gravitational object; are they oscillating to the tune of gravity or is there no correlation.
When we describe time dilation as a result of gravity are we saying that the movement of all the tiny particles including sub atomic particles are actually slowing down ?
 A: It should. In fact this is the core of the principle of relativity and GR is not necessary at this point. To see this, let’s express the principle of relativity in a slightly, but in my opinion, more profound way: “An observer at constant speed should never be able to notice his past acceleration unless he witnesses it by himself”. The idea here is that if someone that was initially at rest relatively to another observer and then goes asleep, only to wake up after one of them was accelerated, it should be impossible for him to say that he was the one that was accelerated. In his frame, both states of constant speed, before and after, should look exactly the same.
Now imagine the classical twin thought experiment. Both twins are initially at rest relatively to each other in outer space. They move into separated spaceships, set a timer to randomly accelerate one of them, and they go into deep sleep. When they wake up, sometime after the acceleration phase, one of them will notice that her’s aging and wrist clock do seem to agree. She was a young girl at the beginning of the experiment, but her grey hairs and wrinkles do agree with the 50 years passed in her clock. Now, if acceleration don’t affect all clocks in the world (including biological ones) the exactly same way, her sister in the other spaceship will be able to say that she was the accelerated one, since her aging don’t match the 1 year pass of her wrist watch, even if at this phase in the experiment she is in a uniform movement. In other words, the girls have invented a way to check their own state of constant motion, which should be always impossible to do, no matter what.
Replace the acceleration phase with a gravitational slingshot and if constant motion should be impossible to check by oneself, you have no choice than to admit that gravity affects all clocks the same way.
___Edit
The key point here is that all kind of imaginable clocks (atomic, mechanical, you name it...) in the same frame measure the same time but those at different frames measure a different value, and this difference can only be tell when clocks that were initially synchronized, are accelerated and then put back together. It is impossible to check this difference from within an individual frame of reference without looking outside. Due to this fact, it is impossible for someone in any individual frame to feel a sort of time dilation. So time dilation is not a physical force that is making the clocks in the accelerated frame to slow down. What one can say, is that the energy put into the system (the one causing the acceleration) fundamentally changes the way those frames relate space and time rates with each other by changing the correction factor (represented by the metric tensor) that they use to interchange spacetime measurements.
A: I feel like I need to add this answer because currently there are two excellent answers by @JohnRennie, and @gandalf61, but if you look at them at first glance, it might seem as they are contradictory. In reality they are not, but I believe it needs a little explanation.
See gandalf61's excellent answer tells you right upfront, yes, oscillating atoms are actually slowed down inside a gravitational field. Then John Rennie's excellent answer says that time dilation is not an active process, it exists rather, because two observers measure time in different ways.

the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.

https://en.wikipedia.org/wiki/Equivalence_principle
Now as impossible it might seem, both are right. It is very important to understand that speed is relative but acceleration is absolute. According to the equivalence principle, the effects of a gravitational field can be indistinguishable from acceleration.
As John Rennie explains it, the two atomic clocks are in two different strength gravitational field, thus they move along different time axes (the effects of the gravitational field bend the axes in different ways if you will), they measure time differently.
In reality, gandalf61 says the same thing in a comment, explaining, that in an experiment, two atomic clocks were flown around and when meeting again (together and with a third clock), their differences (meaning that they ticked different amounts), were in line with GR.
They are saying the same thing, that the clocks were moving along different time axes (partly because they were moving/existing at different depths down a gravitational potential), and one time axis was bent more by the effects of gravity then another, causing the clocks to tick at different rates.
It is very important to understand that this effect on the clock rates is only realizable when comparing it with other clocks, that were existing at different depths down a gravitational potential (were moving along different time axes). If you have a single clock, you cannot tell that the clock starts to tick slower as you move into a stronger gravitational field (potential), because there is nothing to compare it to. You must always compare your clock to another clock that is in another strength gravitational field (is at different depths down a gravitational potential) to realize the clock rate difference.
So as you say, is this effect absolute on the clocks? You could say yes, but that would be misleading, because if there is nothing to compare it to, you cannot realize it. You always have to compare your clock to another to realize the effect of GR time dilation.
A: 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.
A: I believe that only @JohnRennie's answer properly addresses the point of the question. Time dilation is an effect that is apparent only to observers moving relative to the frame that appears to be time-dilated- in that frame itself, nothing changes.
It is often said that moving clocks run slow, which is a misleading statement, since it conveys the impression that the clock itself has slowed, which is not correct.
If you increase your speed then your local time (ie time where you are in your rest frame) passes at the same rate as time at any other point in any other reference frame. It is only when an elapsed time at a place in one frame is compared with the time at two separate places in a moving frame, that a discrepancy appears.
In SR, the phenomenon of time dilation arises because the axis of time in your rest frame is tilted relative to the axis of time in a reference frame moving relative to you. This means that as you move through space in another reference frame, you move from regions of earlier time in that frame to regions of later time.
The analogy is not exact, but if you imagine travelling east around the Earth leaving London at noon, you will pass through successive time zones in which the local time is progressively later. If you land in Tokyo at midnight on your watch, the local time will be 7am, so you will appear to have lost 7 hours. It is not because your watch has run slow, but because you have moved to a location where time is ahead by seven hours.
The phenomenon of time dilation in SR arises in a similar way. As you move through a frame of reference, the baseline for your movement is not a horizontal plane of simultaneity in that frame, but a sloping one. In your rest frame at any given moment it is the same time everywhere. If it is exactly noon where you are, it is also exactly noon a mile ahead of you. However, at the point in spacetime that corresponds to noon a mile ahead of you in your frame  it is already past noon in the moving frame. Just as when you take-off at noon in London it is already 7pm in Tokyo.
If you still need to convince yourself that moving clocks do not actually slow down, consider the following thought experiement. If you glance at the second hand on your watch you will see it moving at a certain rate. Relative to a frame passing you at 1000mph, your watch will appear to be continually losing a small amount of time when compared to successive clocks in that frame. Relative to a frame passing you at 100,000 mph your watch will appear to be running even more slowly. Compared with clocks in a frame passing you at a fast enough speed, your second hand will appear to be ticking once a year. Your watch seems to be running at different rates depending upon the relative speed of the frame against which it is compared. Clearly your watch cannot actually slow down by all those different amounts at once- the reality is that it always ticks at the same rate in your frame, but in other frames, in which the clocks are systematically out of synch with yours, it seems to be running slow.
In GR, time dilation has a similar cause, but rather than involving planes of simultaneity that are tilted relative to each other, it involves curved surfaces of time.
A: 
Are the oscillating atoms actually slowed down by the effect of time dilation

Yes. The Hafele-Keating experiment shows that atomic clocks are affected by gravitational time dilation. And the Pound-Rebka experiment shows that the frequency of gamma rays emitted by excited atoms is affected by gravitational time dilation.
Gravitational time dilation is an actual objective effect, unlike the apparent subjective time dilation of special relativity. Observers who are stationary with respect to each other (in space) but at different “heights” in a gravitational field will agree that time runs at different rates in each of their localities, and the observer for whom time runs at a slower rate is deeper in the gravitational field. Observers who are in the same locality but have different acceleration histories will have different measurements of elapsed time and will agree who has had the greater acceleration.
No observer can detect gravitational time dilation by purely local observations in space. Everyone’s local clocks run at one second per second - this is a tautology. But comparison of non-local clocks that are stationary (in space) with respect to one another establishes that gravitational time dilation is a real effect.
A: The following is rather only a comment on the present OP question (which exceeds 600 characters, however):

Harvey: "[...] Are the oscillating atoms actually slowed down by the effect of time dilation resulting from a greater gravitational object [?]"

Should this question, and the OP question as a whole, be construed in the following concrete sense, for example:
"Is the groundstate hyperfine transition frequency of a caesium-133 atom which is suitably supported against falling (towards some nearby, significantly massive object such as the Earth) smaller than the unperturbed groundstate hyperfine transition frequency of a (suitably free) caesium-133 atom ?"
??
(If so, I'm hardly qualified nor particularly interested to answer this question, or similar questions, about atomic physics.)
If not, then, in my understanding, the OP question statement requires clarification, revision, improvement. (This may obviously be approached in various ways. One suggestion which I'd consider worthwhile might be asking concretely:
"What exactly do we mean by saying that one oscillator had been `oscillating slower´ than some other oscillator ?" ...)
A: Well, yes.   The Pound-Rebka experiment  Pound-Rebka gravity doppler shift
definitely indicates that physical material systems with
characteristic frequencies DO have different behavior depending on their relative positions in a gravity field.   Their
internal clocks disagree, just as General Relativity requires.
The effect is small, but some very accurate gamma ray frequency determination verifies the time variation.
A: Atomic clocks keep track of the regular oscillation of atoms. This is how they are able to keep incredibly accurate time.
The closer an atomic clock is to the source of gravitation, the slower time it records, while the farther away it is, the faster the time it records (as measured by distant observers).
The fact that gravitational fields affect atomic clocks in this way, we can safely assume that the oscillations of atoms is actually slowed down by gravitational fields (as measured by a distant observer) .
A: Quoting the part of your question I'm responding to:

When we describe time dilation as a result of gravity are we saying that the movement of all the tiny particles including sub atomic particles are actually slowing down ?

We have to be careful here with the expression 'slowing down'.
Let me make a comparison:
As we know: heat is motion of the atoms that a substance consists of. The atoms/molecules of a gas have a distribution of velocities. That distribution has an average. The higher the temperature the higher the average velocity. Cooling a substance means you are decreasing the thermal velocity.
Time dilation effect is not comparable to that kind of slowing down.
I strongly recommend against thinking in terms of slower/faster, it will lead you astray.

A more cautious approach is to think of time dilation in terms of an operational description of it.
Physics.SE contributor Gandalf61 already mentioned the Hafele-Keating experiment.
The point in the experiment that the rubber meets the road is when the two clocks are rejoined. When the two clocks are right next to each other again the comparison is without ambiguity.
It is seen that for one clock a smaller amount of proper time has elapsed than for the other clock.
I repeat: I recommend thinking in terms of amount of proper time, and not thinking in terms of slower/faster.
The idea here is to be a minimalist. Express the minimum that suffices to describe the observation, and stay away from interpretation of the observation

For the way that gravity operates no underlying mechanism is known. In order to formulate GR at all the principle of equivalence must be granted.
The justification for granting that assumption is the success of GR as a theory of physics.

In some way shape or form gravity sets up a bias, such that deeper in a gravitational well a smaller amount of proper time elapses as compared to the amount of proper time that elapses at higher altitude.
This bias, whatever its nature is, is a true physical effect. This bias is not an apparent effect.
