In baseball, this position refers to the fourth hitter in the lineup. The cleanup hitter is traditionally the batter in the lineup who provides the most power (home runs, extra-base hits, etc.).
Teams generally try to deliver a "one-two punch" between the third hitter, who usually has the best average, and the cleanup hitter, who has the potential to create a big inning with additional RBIs. This player is essentially "cleaning up" by clearing the bases of the (potentially) three hitters preceding him.