When playing Don't Starve on the Standalone version, save files and game progress are saved only on the machine that the game is being played on. Steam however, allows users to save on the Steam Cloud which then allows them to play their saved games on different computers as long as they are using the same Steam account.
More information on Steam Cloud here: https://help.steampowered.com/en/faqs/view/68D2-35AB-09A9-7678