When Kimmel first started on ABC, his show couldn't book any guests that weren't contracted to ABC. Kimmel obviously didn't have a very large guest list (Adam Corolla appeared a couple times a week), so to make up for it one night he told the audience Matt Damon would be on. Of course, Damon was already contracted to CBS and wasn't actually planned for the show, so Kimmel made the joke that they ran out of time for Matt Damon because he'd spent too much time talking (again) with Adam Corolla. It became a running joke for Kimmel to apologize to Matt Damon at the end of his show for running out of time.
Matt Damon flipping out in this video is a joke. Both Jimmy Kimmel and Matt Damon have said it was planned.