(From a friend one mine; I'm just hosting this here.)
Your buddy Einstein is riding the interstellar express train. All the lights are off, but just as it passes your observation platform, he turns on a lamp in the middle of the train. You and he have taken bets on which end of the train, front or back, will light up first. As expected, your video cameras on the platform record the rear of the train lighting up first, since, as the light spreads out from the original location of the lamp, the rear of the train moves slightly closer to the source, while the front end is moving away. However, Einstein reports that his equipment shows both ends lit up simultaneously.
Both of you are correct. In your reference frame the events "light reaches front of train" and "light reaches rear of train" are not simultaneous, even though in Einstein's they are are.
Now suppose your friend Bob has the ability to teleport instantaneously. Einstein asks him to help out in the experiment by waiting in the front of the train, and teleporting to the back the moment the light reaches him. Then, when the light reaches the back of the train, he'll teleport just out to the side of the train to make sure the situation looks the same in the rear from outside, at which point he'll teleport back up the platform to check on the front of the train, where he'll stick around to discuss the results with you.
Riding on the train, Bob sees the light hit the front, and he teleports to the back, which he notes has also just lit up, confirming Einstein's observations. He teleports onto the platform, where he checks that yes, the rear of the train is lit up, so he teleports to the front of the train... where he confirms your observation the front is still dark.
Also, he notes that there's a copy of him still sitting in the dark waiting for the light to show up.
All this reasoning works with any faster than light travel, rather than just instantaneous teleportation, but takes more math.