The way I've understood it is that everything which can 'move' in spacetime has to have a momentum vector with constant magnitude C (when expressed in the right units, C = 1)
The faster you 'move' through space, the more you have to 'borrow' from the time component of the vector to maintain a magnitude of C
That means your 'position in time' moves slower; and so for people who aren't moving as fast through space as you are, they appear to 'experience more time'
Therein lies the rub. To "move" proves nothing about who is "faster" because there is no absolute frame of reference. You may think that I am moving at 0.8c, but maybe it is just me slowing down to a standstill while you are still receding at 0.8c. This might be a valid interpretation of your observation of me "moving at 0.8c" if... (and only if...) there were an absolute frame of reference. But there ain't.
AIUI it is the flavors of acceleration - including accelerating, decelerating, and gravity - that tinker with time. Which is why I still can't quite wrap my mind around the Twin Paradox, because it is usually explained in terms of speed, not periods of acceleration.
What is the frame of reference, or is there even one, in this case? If you have one spacecraft moving in one direction at 0.5c flying by Earth, is the craft at 0.5c and experiencing "slower" time, or is Earth experiencing slower time? Or do they both experience the same thing as they're not accelerating wrt each other?
If they just pass by each other without changing speeds, each sees the other as experiencing slower time. For the case where one leaves and comes back, see the Twin paradox[1] which is resolved by the simple fact that to change speeds one must accelerate (or that the one who returns must have experienced at a minimum two different frames of references; one to leave and one to return).
There is no independent frame of reference by which we can tell whose time is slower.
From the spacecraft's frame of reference, Earth's time is slower. From Earth's frame of reference, the spacecraft's time is slower. Both are right.
When we say that time is slower or faster on a spacecraft, the Moon, or an exoplanet of Christopher Nolan's, we are implicitly prefixing the statement with "From Earth's frame of reference..."
There isn’t really a “why”, other than the need to match observed reality.
We know from observations that light moves at a constant speed, even when the observer is moving near the speed of light, and we know that this observation is true regardless of your frame of reference.
In order for physics to remain consistent while accounting for the constant speed of light, other things need to flex between the two reference frames: namely, time (time dilation) and length (Lorentz contraction).
The speed of light is a universal constant in a vacuum, like the vacuum of space. However, light can* slow down slightly when it passes through an absorbing medium, like water (225,000 kilometers per second = 140,000 miles per second) or glass (200,000 kilometers per second = 124,000 miles per second).*
Light propagation in a medium is a quite different thing from light in vacuum.
For example, the speed of light in a medium is not the "speed limit" of things in the same medium, and particles in it can actually move faster than light: https://en.wikipedia.org/wiki/Cherenkov_radiation
In other words, the speed of light in vacuum plays a special role in both a vacuum and a medium.
In other words, "to make the math work out". That's kinda what I was poking at, trying to understand if that is some fundamental truth or if it is the result of some underlying mechanism that is more fundamental.
But a century of experimental and observational data proves that it is.
At this point it's generally just taken as a fact that the speed of light is constant for all observers. The explanation given above falls out as a direct mathematical consequence.
The faster you 'move' through space, the more you have to 'borrow' from the time component of the vector to maintain a magnitude of C
That means your 'position in time' moves slower; and so for people who aren't moving as fast through space as you are, they appear to 'experience more time'