Ernie sings about his dream.
Ernie dreamed that he was waking up? It's like Inception! (By the way, how can it be that none of the extended Muppet family has done an Inception parody yet? Get on it, people!)
This song kind of reminds me of "Sit Down, You're Rocking the Boat" from Guys and Dolls, in that they're both songs telling about a dream, and the structure is kind of similar (a slow bit followed by a fast bit, alternating). I wonder if it was specifically meant to be an homage. Content-wise, there doesn't seem to be a connection.
(Yes, I've linked to two different versions of "Sit Down, You're Rocking the Boat" since first posting this, and it's because I can't find a version that's exactly like the one in my head. But this version is probably the closest.)