Conceptually, how does replay work in a game?

I think your initial thought was correct. To create a replay, you store all input received from the user (along with the frame number at which it was received) along with the initial seeds of any random number generators. To replay the game, you reset your PRNGs using the saved seeds and feed the game engine the same sequence of input (synchronized to the frame numbers). Since many games will update the game state based on the amount of time that passes between frames, you may also need to store the length of each frame.

Starcraft and Starcraft: Brood War had a replay feature. After a match was completed, you could choose to save the replay to view later. While replaying, you could scroll around the map and click on units and buildings, but not change their behaviors.

I remember once watching a replay of a match that had been played in the original game, but the replay was being viewed in Brood War. For those unfamiliar, Brood War contains all of the original units and buildings, as well as a variety of new ones. In the original game, the player had defeated the computer by creating units that the computer could not easily counter. When I played the replay in Brood War, the computer had access to different units, which it created and used to defeat the player. So the exact same replay file resulted in a different winner depending on which version of Starcraft was playing the file.

I always found the concept fascinating. It would seem that the replay feature worked by recording all of the inputs of the player, and assumed that the computer would respond to those stimuli in the exact same way each time. When the player inputs were fed into the original Starcraft replayer, the game played out exactly as it did in the original match. When the same exact input was fed into the Brood War replayer, the computer reacted differently, created stronger units, and won the game.

Something to keep in mind if you're writing a replay engine.