What does the phrase "humor me" mean?

Humor, as a verb, means literally to indulge or tolerate someone's humor (noun), where the noun intends not the modern sense of joking or a transient mood but the now archaic sense of temperament or idiosyncracy or eccentricity.

Humor me thus means indulge me—in the sense of gimme some slack or gimme a break, but less aggressive than these. It is used most often as an appeal, at once gentle and ironic, to an interlocutor who interrupts one's discourse; it means, approximately, Let's treat what I'm saying (or doing) and you're objecting to as mere personal whim—on that basis, allow me to finish, and then you can have your say.

"Hey, friend. Can I get you to stand right here under this teetering bucket of water?"

"Why?"

"Humor me."

"Oh, alright."

In my opinion, it isn't necessarily sarcastic, it's simply a way of saying: "Just comply with what I'm saying/doing right now and you can contradict me later."