I don't know if it has been mentioned, but "dying" or "getting killed" are commonly referred to by comedians when jokes fail.
Stewart was condescending, perhaps with reason, to him about the whole having his hands in his pockets thing. Corden is a grown man, if he wants to stand there with his hands in his pockets, it's his business.
After Corden then got pissed and said presenters should get the F*ck on, Stewart tried to insult him on a personal level, which was very un-knightly. But he couldn't finish the punchline, that's when Corden said he'd like to see him try to find one, he'd like to see him die - and Stewart did, because when he finally got a punchline, it bombed.
I'd say neither men handled the situation very well, but the amount of Stewart nuthuggers in this forum is awesome.