Oh, yeah, it's ALL handwavy bulls#!t. It's a 60s sci-fi TV show. A great one, but... you know.

I'll say that the transporters are some of the most consistent pieces of tech they came up with, though, at least as they get explored over time. They need a beam, they are disrupted by shields and interference, they turn people into a data buffer "pattern" that seems to follow the way data would behave, in that they can add and substract to it. You need to assume they don't use them as full-on cloning machines because of regulations, rather than tech limits, but it mostly makes sense.

Unfortunately the version that makes sense is the most disturbing interpretation, so they still need to handwave the crap out of it.