I think the "magic technology" thing is a bit of a mislead. It isn't that Star Trek never had fanciful things in it, it's more that it took the implications of its tech a lot more seriously and thought it all through a lot more than the more "soft" sci fi shows like Star Wars. Like in Star Wars you have FTL travel, sentient robots, laser cannons and stuff - but for some reason everything is either the Wild West or World War II and it's best if you don't think about it too hard. Star Trek though goes way out of its way to explain how its tech shapes its worlds and the people on them, so it's a lot more jarring when the technology is just being made up on the fly instead of based on something that's been preestablished.

Anyway the transporters work by disassembling you at an atomic level and reassembling you somewhere else. It's unclear whether the "you" at the other end of any given teleportation is actually you or if you die in the process, but I'm of the opinion that the clones that sometimes emerge (both evil and not) from transporters imply that the original is being killed every time someone transports but that the galaxy is ideologically committed to telling itself over and over that that isn't the case.