According to Bill Simmons of ESPN.com, a close friend of Jimmy Kimmel, it wasn't staged.
Eric (New York, NY): Please tell us what really happened with Matt Damon on the Jimmy Kimmel show.
Bill Simmons: Jimmy will get mad at me for this, but screw it - Damon was supposed to be the last act of the show, and they screwed up the time and only gave him like 45 seconds, he didn't even get a chance to show a clip from his movie. So Damon totally flipped out, it was like the scene from Good Will Hunting when he flips out on the math professor. I can't believe nobody made a bigger deal about it. Everyone thought it was a setup but it actually happened, Jimmy was pretty shaken afterwards.