Generally they apply through the state party, often because they know someone.

Otherwise it's connections through the polisci programs, the professors help them get their resumes out, or again the state parties do.

For more powerful offices it's mostly the professors at gt, jfk, Yale, etc, then they recruit people they know and trust from home, scions of rich donors who went to the right schools.

For lower power offices and representatives you basically just send your resume and their chief of staff reviews it like any job. They also have a ton of interns, similar process, influential offices use connections, lower offices get whats left. Finally there's the page program, connections help to get in, professors often encourage kids to apply.

For the lower levels a lot of it is energy, the jobs barely pay and they're miserable, so anyone who volunteered on a campaign has a good shot for a freshman congressman whose seat wasn't heavily contested.