What's that between Matt Damon and Jimmy Kimmel? [closed]

People magazine did an article about this "feud" that you can read online. Basically it was a throw away joke that they thought was funny so it became a recurring thing. There are lots more details in the article but the meat of it is here:

“We had a bad show … The guests were bad, and I was feeling pretty bad about myself at the end of the program,” Kimmel told NPR in 2013. “And I decided to say, for the amusement of one of our producers who was standing next to me … ‘I want to apologize to Matt Damon. We ran out of time.’ … And he got a kick out of it, the producer, so I just started doing it every night to amuse him.”

And even though he only said Damon’s name because it’s the first A-lister he thought of, the joke stuck after the December 2005 episode. The host then spent about a year closing his show with the same apology to Damon before the actor himself came on to participate in a fake fight in 2006 after he finally makes it on the show and the host cuts him off.

As a Coda to Erik's answer:

After a while Matt Damon got in on the joke. They'd show clips of him backstage acting all ticked that he was bumped. It sort of snowballed from there. They'd start doing bits where Damon snuck onto the set, or even took over the show. In 2008 Kimmel's then-girlfriend Sarah Silverman got in on the act by presenting on the show an entire song about how she was cheating on him with Damon.

Because we know its not true, its kind of funny. Also, its good publicity for both of them, so they keep doing it. Time.com has a good article up going into more detail, including lots of youtube clips.