How do light cones limit time travel?

Recall the Minkowski metric
dS2 = c2 dT2 - dX2
The value of dS2 is important in Special Relativity because it stays the same under a Lorentz transformation. That means all observers moving at constant velocity with respect to each other will agree on the value of dS2. Spacetime intervals
If two spacetime events are separated by the spacetime interval dS2 = 0, then we say those two points have a lightlike separation. Only a light ray, and nothing else, can connect those two spacetime events.
The path of an observer travelling less than the speed of light satisfies dS2 > 0. Relativists call that a timelike path. When two spacetime events are separated by the interval dS2 > 0, we say those points have a timelike separation.
When two spacetime events are separated by the interval dS2 < 0, we call that a spacelike separation. For events with a spacelike separation, there always exists some moving observer who will say those two events happened at the same time.
To prove that, look at the Minkoski metric. Suppose we have a spacelike separation with dS2 = -1. Then we can find a Lorentz transformed coordinate system T',X' so that dT' = 0, dX' = 1. This is allowable because it leaves dS2 = c2 dT2 - dX2 = -1 unchanged.
The same logic shows that if two spacetime events have a timelike separation, then there is no Lorentz transformation to coordinates in which they occur at the same time. This is because the definition of simultaneous in this context is dT' = 0. That would make dS2 = - dX2 < 0 . But if the interval we're starting with has dS2 > 0 , and any Lorentz transformation will preserve that condition.

What light cones do

The light cone of an event E forms the boundary between all spacetime events with a timelike separation from E and all events with a spacelike separation from E.
Observe in the figure above that the light cone disconnects the past and future timelike areas with areas of spacelike separation in a Lorentz-invariant manner, so that the gap between the past and future of an event is absolute and unbridgable - for all observers.
This is how Special Relativity implements causality and makes nontrivial time travel so mentally challenging to cook up!
The challenge is to find some way to make the past and future light cones of some event bend around or overlap. But we can't do that in Special Relativity, the light cones in Special Relativity are rigid and unchanging. We need to move up to General Relativity if we want to try messing with causality. Stay tuned!