Cagle asked Damon, 50, how he first heard Kimmel end his late-night show with, "Sorry, Matt Damon, we ran out of time."
"People started to call me and ask like, 'Hey, what's your connection to this guy?'" recalled Damon, who said he hadn't watched the show previously. "I'd never met Jimmy. And he literally pulled my name out of thin air one night."
The "Ocean's 11" star said Kimmel told him that one night he had a ventriloquist and a man in a gorilla suit as his guests. And, as a "throwaway," he ended his show with, "My apologies to Matt Damon, we ran out of time." According to Damon, the joke "changed the course of [their] lives."
"He could have said Brad Pitt. He could've said, you know, George [Clooney], he could have said somebody else," Damon explained. "But he just, for some reason -- maybe I had a movie out that week or something -- but he just said my name."
"It changed the course of both of our lives," he continued. "We've kept this feud going for, you know, it's gotta be 15 years now. [I've] had a lot of fun doing it."
Damon pointed out that Kimmel said the running gag "doesn't behave like any joke ever."
"He goes, 'I say the same joke every night ... and it gets a laugh every night,'" Damon explained.
"He goes, 'It's the weirdest thing.'"
Kimmel, 53, has discussed the origins of the long-running joke in the past. While speaking with NPR back in 2013, the late-night host said Damon's name "was just the first name that popped into [his] head" and his producer "got a kick out of it" so he "just started doing it every night to amuse him."
"Repeating the same joke every single night, you'd think eventually people would get tired of it, but they don't," Kimmel said at the time.