Listening to any Tom Waits song, especially these days, does take a bit of effort and attention, but stick through it. It was introduced as a new song, and Waits says, "This is for you, Dave," so it's pretty clearly about Letterman. But what's so wonderful about this is that if you heard it in any other context, you likely think, "Wow, that's a terrific song." IN context, it's masterful.
One note. As a gag earlier on the show, guest George Clooney came on and said how it was wrong that Letterman was leaving, and that he wouldn't let him. So, he then took out a pair of handcuffs and cuffed himself and Letterman together, and then threw the key into the audience. So, when you see them still handcuffed together at the end of the show, that's why.
So, here's the song, take one last look --