The phrase is probably a shortened form of "shut up your mouth" or "shut your mouth up" Its use is generally considered rude and impolite, and may also be considered a form of profanity by some.
See also: Shut Up and Drive, Shut Up and Dance (Walk the Moon song), Shut Up and Dance (Black Mirror), Put Up or Shut Up, Sinking of the Titanic, and Barkley, Shut Up and Jam: Gaiden.