Each night, the transporter can choose 2 players. He can target himself if he wants. He can't target leavers.
He will visit both of his targets, who will receive a message telling them "You were transported to another location", confirming the existence of a transporter or an hypnotist in the game.
Anyone who visits a transported player will unknowingly visit the player who was transported with them, even if that's normally impossible (for example, a transporter can make mafia members visit each other or even themselves when that's something they can't do intentionally).
The transporter has one of the highest priorities, above witches, escorts and consorts, which means his ability works before their's. That's why those roles can be fooled by transporters, while transporters are roleblock and control immune (if some of those roles visit the transporter, he will receive a message saying "someone tried to control/distract you, but you were immune". The visitor won't receive any special notification).
If you try to transport someone in jail your ability will fail and no one will be transported. You will receive a notification saying "Your ability failed because one of your targets was jailed", and the jailee will receive a message saying "Someone tried to transport you but you were in jail".
If 2 transporters swap the same player on the same night, the transporter who joined the lobby first will have a higher priority.
Transporters can't be reanimated by necromancers or retributionists because their UI is too complicated for them (due to the double targets).
Basically, the point of being a transporter is trying to predict who the evils will kill and who is likely evil, so you can swap them and hopefully save and confirm a townie while killing an enemy in the proccess. You can also prevent other visits like roleblocks on controls. If you transport someone with a mafia member, that person will be safe because mafia members can't visit each other, so they won't be able to visit that player unless a witch interferes. Even if you transport evils into killing a different townie, you will still confirm that the other transported player was the original target, clearing them from any suspicion (at least, of being a member of the attacking faction). And unlike vigilantes, you won't suffer any guilt from getting a townie killed, and won't have a limited amount of uses of your ability. You can swap yourself with a suspect every night and mafia will be unable to kill you unless they reach mayority, or you can make a combo with a TP and make it even harder for evils to kill any of you. And since you are the only role besides hypnotist who gives the transported notification, you are one of the easiest roles to confirm, especially outside of coven.
However, this role requires a lot of skill, since it has a very high potential of backfiring. You can mess up with TI results and make vigis shoot townies instead of evils. That's why it's important claiming day 2 and clarifying any confusion, while reminding any vigilante to tell you who they are shooting so you don't transport them. You should also whisper the jailor so they tell you who they are jailing, so your ability doesn't fail.
If you want more information, check the wiki page about it:
https://town-of-salem.fandom.com/wiki/TransporterI made a post with a simple but effective strat for transporters too, if you want to read it:
https://www.blankmediagames.com/phpbb/viewtopic.php?f=3&t=111824Hope this was helpful.