Why does every Bollywood movie have an unrelated dance and song number?

No idea. I think it's just become tradition. They don't do it in really serious movies.

But if you ever get the chance you must go see a Bollywood movie in India. It's a great time. Saw one in this theater. Huge and everyone is in to it, but not in an annoying way.

Drama (as in dramatic rendition, not the genre) develops differently in different cultures. Not every culture develops drama as English-speaking cultures did (and even in England during the Middle Ages, various types of drama were highly stylized. You could travel across England attending passion plays, for instance, and pretty much know exactly what to expect in terms of characters and symbolism.) There are any number of other examples: Greek chorus style of, well, Greece (strophe, antistrophe, and epode of Grecian tragedy); kabuki from Japan, which is extremely stylized in performance, dance, and make-up (the modern ninja, for instance, came from kabuki using dressing ninjas in that fashion, after the dress of stage workers, to indicate that they 'weren't visible' even when the audience could clearly see them.) Central African cultures make extensive use of iconized masquerade. And so forth. Why Indian drama evolved like that is less important than the fact that it performs the function of a cultural touchstone. The very fact that you can identify a Bollywood production thusly allows Bollywood maintain a certain hermetic fidelity within the context of Indian culture. It doesn't appeal to most Westerners, granted, but, then, is it really all that different from the convention of a character in a musical singing about his or her feelings while every other character pretends not to hear it?