Respawn points are invisible spots located throughout each map that set where you will respawn to if you void out, fall in lava, or fall out of bounds. Respawn points are triggered simply by walking over them.
Respawn manipulation is the method of transporting one or more players' respawn positions in an unintended manner, either by only walking over what is considered "dangerous territory" by the game (such as the edges of land, temporary land, or moving platforms) to dodge respawn points, and/or transferring the respawn position to other players/doppels by toteming them. Respawn manipulation can be used to get all three players to locations that only one player has reached in order to skip parts of a stage.
Example visualization of how respawn positions move around with the players:
A practical use of respawn manipulation to skip the obstacle course in Ice Temple 3: