The underlying reason is causality. Causality is the concept that an event can only be caused by another event in its past (not its future), and events can only impact their own future. Causality cannot propagate faster than the speed of light, ergo for a given event you can express the area of space that could have caused that event as a function of time (it's a circle with radius equal to the amount of time in the past you're looking times the speed of light; e.g. 10s before an event, that event could have been caused by another event anywhere within 10s*the speed of light).

If time is constant, objects moving at a significant fraction of the speed of light break causality. They're moving fast enough that causality propagating the opposite direction of their velocity (things happening in the future) reach them faster than they should, and causality propagating in the same direction as them reaches them slower than it should (because they're moving away from it at a significant portion of the speed causality is approaching them at).

E.g. lets say there are 2 massive celestial bodies A and B. A is not moving at all, B is moving at 50% the speed of light. Let's say they pass close enough to gravitationally interact, but are half a light-year away from each other. Once they pass each other, causality from B will propagate to A at the speed of light, like normal. Causality seems fine.

But causality will propagate from A to B much, much slower because of the relative velocity. If B is 1 light-year away, it would actually take something like 1.3 years for causality to reach it. In other words, A is within B's causality radius (B can cause effects on A), but B is not within A's causality radius (or rather, it is when the event happens, but it won't be there by the time causality gets there). That's a problem for something like gravity. B's gravity can influence A, but A's gravity can't be the cause of events on B, because of the speed of causality. Thus the only valid event is B's gravity on A, accelerating A without decelerating B, meaning we would have actually created energy (at least until causality catches up).

To maintain causality, and preservation of energy, something has to happen to B such that A interacts with B at the same time B interacts with A. The answer is to make movement through time and movement through space inversely correlated. If B is moving fast enough that light takes 30% longer to get there, B's time has to slow down by the same amount so that causality can be simultaneous and not create energy.

Causality essentially requires movement through space and movement through time to add up to some constant. As movement through space increases, movement through time decreases and vice versa. It's basically a formula like (current speed/speed of light) + rate of passage of time = 1.

That's the underpinnings of the idea that FTL travel will allow time travel. If (current speed/speed of light) is greater than 1, the passage of time has to be negative or flowing backwards to maintain causality. I.e. you are moving so quickly that you can catch up to and interact with causality propagating through the universe.

Somewhere out on the edges of the universe, the causality of the meteor that killed the dinos is still propagating outwards. Perhaps if we could move fast enough to reach that wave of causality, we would be able to interact with it somehow. It's all theoretical, and hand-wavy, and trippy, but interesting in concept.