How does Time Dilation Really Work? I've been trying to translate the idea of Time Dilation into something a little more palatable to understand to the lay-person. The light-clock idea is plain and simple and makes sense to me (though as a little side question I'm still not sure whether it's suggesting that the registration of the events - or light itself - is what defines time in that example, beyond light's usage as a clock specifically).
But I still don't understand how this alters the experience (edit: physical experience*) of the individual. The light-clock example, as I interpret it, is more how to an observer, someone else's time can appear to run differently, but I don't understand how that person's time actually runs differently.
I was reading Arthur Eddington's Book The Nature of the Physical World, and as far as I interpret it, he explained that as an object moves closer to the speed of light, it has more energy, and thus gains more mass. And I believe he was suggesting that the increased mass meant that the object required even more energy to move itself faster, including the constituent atoms that form its bodily processes.
So do the atoms, thoughts, and/or bodily functions of a person travelling ever-nearer the speed of light physically slow down relative to their speeds at "rest" on earth as their masses increase in response to their increased speeds?
I'm not at all scientifically literate, mathematically or otherwise, but I feel like there must be some simple common-sense way to understand it, thank you for your help in advance!
(EDIT: I apologise for the bombardment of follow up questions, I'm just really curious about it and I find most answers to the question somewhat circular i.e. Why does time slow down? "Because time slows down." in some form - which seems to permeate itself pretty consistently into popular explanations)
 A: What you need to understand is that the speed of light is not just the speed of light, it is the speed of causality. That is to say, it is the time it takes for two objects separated in space to affect each other.
Light is the mediator of the electro-magnetic force; the attraction/repulsion between two electrically charged particles occurs through the exchange of photons. So two electrons will be repelled from each other with a delay given by their distance apart divided by $C$.
But almost everything in our daily life is governed by the electromagnetic force. Electrons are held to atoms, atoms form molecules, molecules react with each other, materials expand and compress. All of these things are mostly driven by the attraction and repulsion of charges.
Time dilation in a light clock is observed because photons have further to travel and therefore the clock ticks slower; the ends of the clock look further apart (on average) because they are moving and the light has to catch-up. But this doesn't just apply to the clock. It applies to all of the other things I mentioned: atoms, molecules, springs, etc. The photons interacting with an electron in a orbit around an atom will experience the same apparent slowdown as the photon in the light clock.
Furthermore, if we look at other forces (the strong force that holds the nucleus together and gravity) these also work at the speed of light, as far as we know, and are affected by the same time dilation affect. The forces carriers have further to travel and so must work slower.
This is all relative. If I look at a moving light clock (in, say, a spaceship) I see it and everything in its frame ticking slower. But someone travelling in the ship with the clock will see the clock not moving. They will not see any change in the rate of time around them because everything is stationary as far as they are concerned.
A: 
But I still don't understand how this alters the experience of the individual.

In Special Relativity, it doesn't. The only time relativity impacts the experiences of an individual is if they fall into a black hole. Or if they are expecting a relatively moving observer's clock to tick at the same rate and are surprised that it doesn't.
The Lorentz transformations and its consequences only apply when relatively moving frames are compared. If someone is moving by you, with constant velocity and near the speed of light, they don't notice anything unusual in their own frame of references. And you don't notice anything unusual in yours.

So do the atoms, thoughts, and/or bodily functions of a person travelling ever-nearer the speed of light physically slow down relative to their speeds at "rest" on earth as their masses increase in response to their increased speeds?

"Relative to", yes. But, in their own frame, no.
The fundamental seemingly paradoxical thing is that the speed of light is the same in all reference frames. From this, all the other bizzare consequences follow. The equation $c^2 dt^2= dx^2 + dy^2 + dz^2$ expresses this fact and it is preserved by a "rotation" in spacetime in a way that is mathematically similar to how $dx^2 + dy^2 +dz^2$ is preserved by a rotation in space. That is the origin of spacetime and the lorentz transformations.
A: I'll comment on the bit in your question referring to "increasing mass". You can find my answer on time dilation near the end of this post.
To the first bit: in special relativity,
$$E = \gamma mc^2$$
where $E$ is the energy of the body in question, $\ v$ is the speed of the body, $c$ is the speed of light and $\gamma$ is the Lorentz factor.
$$\gamma = 1/\sqrt{(1-v^2/c^2)}$$ In popular culture, people say Einstein envisioned $E = mc^2$ assuming that mass increases if that body's velocity increased.
However, the mass doesn't change at all. I don't know why Eddington explained it otherwise.
If a body moves close to the speed of light, $v \to c$,  the energy shoots up with $\gamma > 1$. If the body actually travels at the speed of light, $\gamma$ becomes infinite, and the body must have infinite energy. Obviously this isn't possible since energy's not unlimited in the universe.
If the body were instead moving at the every day speed of humans, cars, planes, rockets and planets, the Lorentz factor $\gamma$ would be pretty much $1$. Physicists tend to hate the idea that mass is variable, since it's not intuitive where that mass would even come from. From energy? How so?
Regarding time dilation, yes, time actually does slow down in another reference frame if their speeds were approaching that of light. This means, even bodily motion slows down. But this doesn't mean the person will die, since in their reference frame, their cells absorb nutrients and produce energy within like the rest of the world was slowing down.
It's all relative, that's why it's called relativity.
A: 
But I still don't understand how this alters the experience of the individual.

I agree with jelly ears it doesn't. The time dilation is comparison problem, not experience problem. I think people focus too much on time dilation as it is very cool, but they should be focusing on relativity of simultaneity
Imagine scenario when I start my clocks just as my friend is passing by me with relativistic speed and he does the same. I then wait until his clocks tick one second and look at my clocks. My clocks will show more time.
But from the point of view of my friend, it is me that was passing by with relativistic speed and when he looks at his clocks just the moment my clocks say 1 second, he will see his clocks showing more time.
So what gives? How do both clocks tick slower than the other?
The issue is that my friend thinks he is looking at his clocks at the same time as my clocks ticked 1s. But I disagree with him, this is not at all happening at the same time from my point of view. I thus disagree with him, that his time measurement is a correct way to measure time. From my point of view, as my clocks ticked 1s he was still keeping his clocks going and he did not stopped them! No wonder he gets such a weird result.
This is of course weird behavior. Everyone disagrees with everyone about what it means to measure time. Physicists resolved this problem with simple idea - everyone correctly measures time, it just depends on the point of view. But for these measurements to be useful we need some way to switch between different point of views. We do this using lorentz transformations of which time dilation is just special case. In fact, I think we could do without the concept of time dilation altogether, because it ignores/hides the crucial component of relativity of simultaneity and thus leads to a lot of misunderstanding. We could just use lorentz transformations in their entirety and I think all would be a little clearer, albeit less cool.
