I understand the explanation, for example, about a light beam reflecting between two mirrors on the space craft. But what I am looking for is what aspect of traveling at very high velocities must slow down every single process that is time dependent.
Is it anything like the inertia of every particle increases so in some sense the it takes more energy to cause, for example, a chemical reaction to occur?
Time dilation affects, but is not a consequence of, any processes, microscopic or macroscopic. It is a consequence of the Minkowskian geometry of the spacetime in which all processes occur and has absolutely nothing to do with the details of their dynamics.
It's just a direct consequence of the geometry of spacetime. This question is similar to asking,
"What aspect of traveling on the surface of a globe makes you return to your starting point after traveling long enough in a particular direction?"
It's simply the fact that moving along the curved surface eventually changes your direction of travel in such a way that you end up back where you started.
The same is true of Minkowski (i.e. "flat") spacetime - the fact that events in fast-moving systems appear to occur more slowly is a consequence of the fact that the "distance between events" is modified compared to a Euclidean spacetime; moving through this geometry changes the apparent internal motion of the system, just like moving across a globe changes your direction of travel.
You have asked a very central question, and yet one that physics really can't answer. Physics describes how the universe works. The reason why one law is true can only be that it is a consequence of some simpler law. The simplest laws (and many more complex laws) are known to be true because they have been experimentally verified. Nobody knows why the universe works the way it does.
Special relativity is traditionally taught by taking the constant speed of light as the fundamental starting point. The reason for this approach is that it only requires one assumption that violates everyday experience, and it quickly leads to the correct laws.
But it leaves one feeling unsatisfied. Given one confusing, strange but true fact, a lot of other confusing, strange but true laws can be derived from it. One wants to ask "OK, you can't go faster than light, but why not? What if you could?" It is possible to introduce the strangeness another way that might me more satisfying, though no less counter-intuitive.
Time is a lot more like space than you would expect. We don't see this because we move so slowly.
It isn't that many independent processes are all changed the same amount. Time is the thing that changes.
Light travels at $3 \times 10^8$ m/s. We are comfortable at $3$ m/s. We have difficulty understanding relativistic physics.
Consider a world where the fastest motion is $3 \times 10^{-8}$ m/s. This is about 1 m/year, the speed of a glacier. It is not much faster than $1$ cm/year, the speed of continental drift. We can learn about our difficulties by looking at the difficulties glacier world physicists have with everyday physics.
In classical glacier world physics, each object has a fixed, intrinsic property called position. Every observer agrees on the position of a given object. Position can be used as the identity of the object.
However, precise measurements or measurements over long time intervals show that position changes with time. This leads to the counter-intuitive concepts of "velocity" and the "failure of sameplaceity".
These can usually be ignored. But observers traveling at everydayistic velocities would see strange effects. Bob and Alice both agree that they both have position $0$ at time $t_0$. At $t_1$, Bob says he has position $x_0$, just as one would expect. Likewise, Alice says all is normal with her. But Bob says Alice is at $x_1$ and Alice says Bob is at $-x_1$.
This is not only counter-intuitive. It is a paradox. Not only does an object have different positions at different times, but different observers don't agree on what the positions are. They see a difference growing in opposite directions.
Our conceptual difficulties are much the same. We are used to different positions at different times, and two observers measuring different positions for the same object.
We think of position and time as the identify of an event. We think of all events at a given time as identifying the state of the universe. We think all observers should agree on the time measurement of a given event, even if they disagree on position. We do not expect the velocity of the observer to affect the measurement of time.
This understanding of time is incorrect. Instead, we see on a space-time diagram that one observer's present includes another observer's past and future in a velocity dependent way. This leads to all the counter-intuitive concepts and paradoxes of special relativity.
Length contraction is one example. An observer sees a rod at rest. He measures the position of the two ends at the same time. From this he gets the length.
Another moving observer measures the positions of the two ends at his version of the same time. Since he is using different times than the first observer, he is seeing the ends in different places. It is not surprising that he gets a different length.
A particular barrier to understanding time are the concepts of past, present, and future, and how they relate to causality.
Perhaps the hardest misconceptions to get over are these. We think of the present as the only time that exists in the same way that a glacier world observer might think of his position as the only place that exists. Classically, everything in the past can affect everything in the present. The past is gone. The future hasn't happened yet. It hasn't even been determined yet.
Physics is a layered subject. We begin with simple models like classical physics with point particles and forces. We then move on to more complex models like special relativity and QM. If we just consider special relativity and ignore quantum mechanics, future states of the universe are determined by past states. Even if it is counter-intuitive, we can form a self consistent model where past and future are do exist. They are simply elsewhere (or maybe elsewhen?). My past and future are inaccessible to me, but they accessible to other observers moving relative to me, as shown on a space-time diagram. This is something like other places being inaccessible to a glacier world object, even though they are accessible to different objects.
With quantum mechanics, this need not change. The universe is not deterministic in non-relativistic quantum mechanics. Particles and forces are replaced with wave functions and potentials. Future wave functions are calculated from the present wave function with the Schrodinger equation, and future expectation values of measurements are calculated with the future wave functions.
Relativistic quantum mechanics is a much the same. The Schrodinger equation becomes the Dirac equation. The world line of a particle moving into the future becomes something fatter for a wave function moving into the future. For one observer, the present is a snapshot of the wave function at a given time. A moving observer would see a present that includes some past and future of the first observer. Both observers could predict expectation values of future measurements, but they would be somewhat different futures.
