The clip is pretty short, and mostly discussed in the article. But, to cover it:
Letterman is berating Bieber for having tattoos. He asks if he really thinks it helps him.
Bieber talks about getting it since he had a new album coming out, and that it’s not such a big deal (“it doesn’t not help me”).
More discussion of what his parents think; Bieber mentions that his dad has one, and Dave says, “Oh.”
Then Dave leans over and sort of playfully attacks him, trying to ‘rub off’ the new one on Bieber’s arm. Bieber is shouting “Hey! Hey!” but it sounds a bit strange, more like “Eh! Eh!”
Dave says, “You were just going ‘Eh! Eh!’” (making fun of the sound) and Bieber says it’s a Canadian thing.
Then Dave goes into the, “don’t go crazy” bit and the Sistine/16th exchange occurs. After the audience stops laughing at Bieber (who is staring somewhat blankly - he seems not quite sure of what just happened), Dave says “Canadian high schools.”
Dave and Bieber share a fist bump to resolve all differences.