good posts above...takes time and some reflection to understand what they mean...
Time slowing for a fast moving observer can be thought of in many different ways...there
are a lot of fascinating and not obvious effects. That's why it took an 'Einstein' [a smart fellow] to discover them...and he developed his insights before most of the math, somehow understanding things were different than almost all other scientists believed at the time.
When George posts
...we mean that the clocks traveling in that rocket are running at a low rate...
he's referring to the ticking of the clock [passing of time] in the rocket as seen by an outside observer; inside the rocket an observer sees her local time in the rocket tick by at the normal rate, but observes the outside observer's clock ticking slow! In this example from special relativity, different speed observers usually don't agree on the passage of time.
the documentary says is that time itself slows down to prevent the maximum speed, c, to be broken.
I guess that's ok as a start, but misses a lot of what's really happening. Keep in mind we think we understand WHAT is happening[confirmed by many experimental observations] and that's what we model mathematically. We don't fully understand WHY it happens.
For unknown reasons, so far, everyone observes light when measured in flat spacetime as 'c'. [It gets a bit more complicated if gravity is present.] Einstein realized this means that instead of space and time being fixed and immutatable, they are not: it is the speed of light that is constant [fixed and immutable].
In other words, when you travel faster and faster your measures of distant space and time change, those measures locally [right where you are, in your own frame], are not affected: it's called space contraction and time dilation in relativity. Two things affect the passage of time, that is the tick rate of a clock when viewed from a distance: relative speed and gravitational potential.
...how would time (in the perspective of light) behave?
Our mathematics doesn't, strickly speaking, cover that as posted by others. That particular question, as noted above, has no 'real' meaning in relativity. It's kind of like asking "Is red a happy color?" But a rough, crude, answer is that "a photon doesn't age". This
means that when viewed from an outside inertial frame, IF a clock could be carried along at lightspeed, it is believed it would be observed to stop ticking. We know this slowing occurs as velocity inscreases because some fast moving decay particles have been observed to last longer than expected..if they were stationary...their 'half lives' [life times] are extended beyond what is expected as we observe them whizzing by.
