I'm relatively new to speed running the game, and I have heard a lot of talk about frame rules and how they influence the game, in particular, 8-2. I have heard Darbian explain it as if it were a bus leaving a station every 21 frames to go to the next level, but I don't really get that explanation.

Can I get a plain English definition of what frame rules are, and how they precisely influence the game?

There is a timer that counts down from 20 to 0 (and goes back to 20 after 0) every frame. After you finish a level, the game waits for the timer to go to 0 before loading the next stage. So as long as you finish within the same 21-frames-cycle you will go to the next level at the exact same time.