So what even is a Replay File, how do I get one, and why might I even want a Replay File?
A Replay File is a file that records exactly what happened during a game. It has two main components: the starting game state and a list of timestamped moves, or more precisely, moves annotated with the tick number they were executed. This is comparable to a PGN file in chess. We feed the Replay File to this Replayer to replay the game by playing out all the moves from the starting position. Because the physics is deterministic, we can faithfully resimulate the entire game, exactly how the game server simulated it.
Whenever a game ends, the post-game popup gives you both a link to view a saved replay of the game and a button to download the Replay File. Additionally, if you have a registered account, you can access your replays from your publicly viewable profile as well. That said, there are no guarantees that replays will be stored indefinitely. So consider downloading Replay Files for replays you don't want to lose.