What does the phrase "put it down on me" mean?
That song appears to have a dual meaning of the phrase.
we’ll buy another round and it’s all on me
as long as I’m around put it down on me
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.
girl all I really want is you down on me
put it down on me