To add a bit of detail, it comes down to circles being nice, simple geometric objects, and an assembly of metal with contact points being capable of way more accuracy than you'd first expect.
Bootstrapping the first lathe is harder; most likely some historical elite master craftsman was able to make one freehand, and future ones derived from it. We still have the one Vaucanson made that way, although it sounds like it was a one-off. David Gingery wrote a book on the topic, but he still assumes you have a power drill and a ready-made threaded rod.