While playing a First Person Shooter the Cleaning Crew is a player(s) who takes high ground (ex: roof top) and snipes out enemies that are closing in behind the player(s) on the ground. The ground player either missed these enemies the first time though or the enemies just spawn after the player passes a certain point.
Dave: Man, I hate shooter games! The enemies always sneak up and kill me from behind.
Tom: Dude, you just need a good cleaning crew to watch your back.