Well, most of the humor centers around Papageno, the bird-catcher who helps the hero on his quest. He's rather funny, though it's hard to describe the humor. Our Papageno is quite visual, plus, he's dressed as a bird the entire time.
He's looking for a wife, which tends to get humorous, as such things do. You might say that the main storyline is an epic quest, while Papageno's story is more of a comedy.
And the music itself is quite funny. Mozart was a very funny man, after all.