The Reapers have been around for a long, long time. Longer than the Protheans. They hibernate in starless spaces in the galaxies, before returning to harvest all sentient life. Why do they harvest sentient life? We have no idea. Maybe because it’s awesome and evil.
The actual reason is a little more confusing. Potentially as a reproductive cycle to build new Reapers (out of - get this - liquefied life forms). Partly to steal technology developed by the now reap’d races. Partly because they really are just that rude.
They don’t just look like organic ships, either. The Reaper Vanguard (the Reaper that doesn’t hibernate with the rest, instead waiting for the ideal time to signal the return of the other Reapers) was actually used by Saren as his ship.
It started turning him and impairing his ability to make decisions that weren’t “betray the galaxy.” That Reaper’s name was Sovereign, and he was a real jerk. He also looked like a giant Reaper Cuttlefish.
The Reapers have thought this out. They’ve left Mass Relays (like slingshots for space ships) all over the galaxy, and these mass relays all point toward the mysterious, functioning Citadel. That way organic life bases their civilization around the Citadel.