In such a puzzle, the solver is initially given a certain amount of time to complete the puzzle ... usually less time than required.
The Stopwatch adds exactly one minute to the amount of time the solver has remaining to solve the puzzle.
See CheckPointTrigger for information about other ways the solver may earn additional time.
These may sometimez be acquired via a SecretTeleporterTrigger which generatez a Red Warp
(RW) or via a Secret Switch 
, which temporarily changez Tilez, allowing a Grunt to bypass an otherwise impossible barrier.