From the song we also have
Nobody ever loved me like she does
Oh, she does, yeah, she does
And if somebody loved me like she do me
Oh, she do me, yes, she does
I'm in love for the first time
Don't you know it's gonna last
It's a love that lasts forever
It's a love that had no past
So the passage you quoted is a play on two ideas:
- done me as a slang way of talking about sex
- A non-standard way of saying she caused him to fall in love
We know it's just not slang for sex because of this line:
I guess nobody ever really done me
Which means that nobody has ever made him feel real love before.
Given the song title of "Don't let me down" you get that the song is talking about the vulnerability of falling in love. He is afraid of the heartbreak that can follow falling in love.
This phrase can mean different things based on context. "He/She really done me" or "He/She really done/did me good" can also mean someone took advantage of you. This might just be regional US slang, though.