I've read a good number of FTL equals time travel questions and answers both here and on other sites, but for the life of me I just do not get the connection.
Let's take a really simple case: say at 12:00 I teleport to some place 30 light minutes away from Earth. If I then pull out a telescope and take a look at Earth I will see the past from 30-ish minutes ago, but if I then teleport back to Earth I'll be back at 12:01 or whatever. I'm not going to end up in the past just because I was looking at it a minute ago. And if I go to a place where time flows differently due to gravity or some other time shenanigans there might be a mismatch, such as spending only 10 minutes there but returning to Earth to find years have passed (or vice versa) but either way it'll still always be in some sort of future.
I read several explanations involving "light cones", still don't quite get what they are (so if you bring them up, imagine you're talking to small child) or why should I even care about them, if I have a teleporter then the light comping from an object is not the most up-to date information I can get about it.
Several articles also brought up a "light on a train" thought experiment where a light is flashed in the middle of a moving train and is picked up by light detectors, and a stationary observer and an observer on a train moving in opposite direction will see the detectors going off in different order which is supposed to prove something about different frames of reference. But the same thought experiment can be done with sound instead of light and it would have the same results, but as far as I know it's quite possible to break the sound barrier without time travel.
Anyway so the question is: assuming a magical teleporter exists is there any logical reason that it has to lead to time travel? And if so, what is the exact mechanism?