What does the phrase "put it down on me" mean?
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That song appears to have a dual meaning of the phrase.
Implies "I'll pay for it". Put it down (Put the bill) on me (on my tab) so to speak.
"Have a round on the house" means the bar is giving a free drink. If someone says "Have a drink, it's on me" is someone offering a free drink.
However the song also implies a more sexual meaning as well.
It doesn't really mean much of anything by itself, but in the context of the song, it is a vague double entendre for oral sex.
I think a careful reading of these particular lyrics will show that this song is about sodomy, with the "it" in the phrase referring to Shawty's rear end.