Basically, Hader’s seeing some of the jokes for the first time as he’s on-air reading them.
Knowing this makes the bits 10 times funnier to me.
I had assumed this is what was happening. His reaction is way more fun than Fallon’s was (the only other SNL actor I remember breaking often on camera). Fallon’s reaction was like he was in on an inside joke that was being played out at someone else’s expense. Hader’s is straight up delight in the ridiculousness of what Stefon is saying.