How are future Olympic athletes discovered/chosen?

pretty simple, you start doing something, let’s take swimming for example, you get good, you start doing competitions, you get first place a lot, you get noticed, you get signed, you get into the olympics

that’s it in a nutshell

There’s a governing body for each sport that oversees the selection of the team for that sport. That body will organize a qualifying competition shortly before each Olympic games. The top finishers of that competition will then be vetted by the governing body to assess their readiness to compete in the Games. They have the final say over who’s on the team.

So strictly speaking, they’re not “discovered”. They present themselves at the trials, and if they’re the best, they’re in.

Here’s a good synopsis of the process by which the diving team is chosen