It's a really good question, and I don't think anybody has a complete answer. But the best hint I can give you is that I bet you've got the lyrics to Poor Unfortunate Souls memorized -- or if not you, then ask a female friend or relative (who is more the target audience for the film).
Music sticks in the mind, especially a song like that with a beat that drives it forward. Being memorable isn't the same as being convincing, but it does hint at the way it gets in your brain via back doors.
There's more to it than that, of course. One of the most important tools to being convincing in a film is that the other characters accept it. The song allows no interruptions, and so you follow along with Ariel as she does. It's a good confidence technique, actually, and there are ways to apply similar lessons in real life.
That doesn't completely answer it, and I think some of it has to be chalked up to the rules of musical theater. Those are purely convention, but you accept them because they move the plot along. Every genre has that, and they look weird if you're in on them. (I find superhero movies bizarre, but they're perfectly coherent in their internal logic.)