There's a few levels to it. The first is the idea of a Planet of the Apes musical being done at all is pretty ridiculous. By the time the episode came out the movie would have been nearly thirty years old, the most recent movie in the franchise would have been around twenty years old and the most recent tv show would have been dead for a similar time.
It's like making a musical based on Starship Troopers. McClure treating it as his big comeback just emphasizes how pathetic he is.
The low quality of the musical is the next layer. The song is a ripoff of Amadeus, the actors are awful, the lead in to the song is hamfisted, there's an ape dressed as a sexy nurse, they have shit coreography interspersed with break dancing from one guy, and a handful of dumb jokes.
Homer and Bart loving the musical is a show of how much you're supposed to find the thing ridiculous.
Then the final sequence in the show. Very light and happy music while the astronaut finds out the human race destroyed itself. The ape proudly parade past him as he joins them in song, quickly blowing over that fact in time for him to declare his love for Dr. Zaius. The tone is all over the place.
Basically it's funny because the whole production is shit but no one seems to understand that in the show, so it's ridiculous in both concept and execution.