MAN, THAT THING YOU GOT BEHIND YOU IS AMAZING= MAN is just a colloquial way of addressing a boy or a man (other possibilities are GUY in AmE and MATE in BrE, but MAN sounds more fashionable). It can be used when talking to girls too, e.g. "hi man, how you doin’"(= hello, how are you?). It can also be used as an expression to signal you're going to say something. In this case you can use it either with boys or girls (e.g. "Man, this is great"").
THAT THING YOU GOT BEHIND= he means the lowest part of her back, the place where you sit on, your bottom.
AMAZING= superb, fantastic, great, incredible (he obviously likes it)

LET IT RAIN= let it rain is a rap expression for throwing money in the air and having it rain down, so it’s a metaphor for spending a lot of money, that is, having a great and luxurious time, "living la vida loca".

I KNOW YOU GOT A MAN= I know you’ve got a boyfriend.

REPORTED SPEECH
----------------------------
He’s got a few sentences here grammatically incorrect. Well, many of the sentences in this song are grammatically incorrect, but the way he uses reported speech (when you say what somebody said) is very incorrect. Nevertheless, this way of speaking is very common in conversational English, because you are not too worried about grammar correction, but about meaning.
He says, "this is what you should say", and now we would expect direct speech (the actual words spoken), for instance
- this is what you should say: "I’m leaving"
But instead, we find another sentence:
- this is what you should say. Why don’t you tell him that...
The suggestion "why don’t you tell him that..." should be followed by reported speech. For example:
- why don’t you tell him that you are leaving?
But now we find direct speech instead:
- why don’t you tell him that... "I’m leaving"
A correct possibility would be:
- why don’t you tell him: "I’m leaving"
CORRECT ALTERNATIVES:
- this is what you should say: "I’m leaving, I found somebody who..."
- why don’t you tell him: "I’m leaving, I found somebody who..."
- why don’t you tell him that you’re leaving and that you found somebody who...

NO MORE THEM GRAY SKIES= THEM is a colloquial (or dialectal) way of saying THOSE. GRAY is an alternative spelling for GREY (the colour). THOSE GREY SKIES is a metaphor for bad times, an unpleasant situation.

WE FLYING ON A G5= "We are flying on a G5". A G5 is the Gulfstream G500 business jet, a luxury private plane, so it means that we are having a wonderful great time together, a bit like the expression "let it rain" (all very materialistic and monetary, by the way). Notice that when we’re talking about public means of transport (boat, ship, bus, train, plane, etc) we use the preposition ON (not "in").

SHAWTY= a sexy girl. A term originated in Atlanta but now spreading. Especially used as a way of addressing that girl (the same as we saw MAN before, e.g. "hi shawty, how you doin’?")

FLY= cool, sexy (e.g. "that girl is so fly!")

THE ONE THAT KEEP YOU HIGH= it should be KEEPS, because it’s 3rd person singular. To be high is to feel great, fantastic. To keep you high is to make you feel great and continue feeling great.

HAVE ME SINGING ALL NIGHT "OH OH OH"= You have me singing all night because I’m so happy... or it might be a sex metaphor

MAN, SHE GONNA SING= Man, she’s going to sing. MAN is the same as we saw before. SING OH OH OH is the same as we saw before.

IF I TALK IT OUT I'LL WALK IT OUT= If I say it, I'll do it (I mean what I say)

I’LL PUT MY MONEY WHERE MY MOUTH IS= I’ll do as I say, I’ll back up my words with actions (same as the previous sentence)

BADDEST= In standard English, BAD is the opposite of GOOD. In colloquial AmE (slang), BAD may be the opposite of BAD, so it can mean GOOD. In this case, the comparative and superlative is "badder, baddest", and not usually "worse, worst". So, as Michael Jackson said in his song, "I’m bad", means "I’m good, I’m cool". BADDEST means "great, the best".

LITTLE THING= "little" here is not describing size, but showing affection (a very common use of "little")

I’MA ASK YOU= I’m going to ask you. This form of the future is still very colloquial, but getting more and more common, especially in songs. (AmE)

DON’T STRESS= don’t get stressed

TELL HIM "TO THE LEFT"= tell him to get out of your way, to get our of your life. Put him out of your head.

CAUSE= because

WE GONE= we are gone, we left