The description of the "Cleanup" badge simply states that is awarded upon your "First rollback." What is a rollback?
A rollback involves reversing an edit (or multiple edits) on a post to a previous version.
All posts edited outside of a (currently) 5 minute window will display an "edited X time ago" link to the left of the post author's badge. If you click that link, you will see a history of edits to that post. One of the options, for all but the last edit, is "rollback."
Note: those are two separate options,