TELL ME ABOUT IT= We use this phrase to show that we totally agree with what the other person said:
- This work is so hard, I really need a rest.
- Yeah, tell me about it! (= I totally agree, I also feel really tired)
In this song, when Dany sees Sandy’s new look, he exclaims “Sandy!”, which of course means, “you look sooooo cool in that leather thing”. So Sandy says “yea, you’re right, I look so damn cool, I know”.

STUD= (coll. AmE) A man regarded as virile, attractive and probably sexually active. But really, this word is often used just as an equivalent of “man”, to address a boy/man:
- Hi man, what’s up? = Hi stud, how you doin’?

I GOT CHILLS= I’ve got shivers down my spine. 
If you have chills, you feel like an electrical charge running up your body, like when you suddenly feel cold or very shocked or excited about something.

MULTIPLYING= If something multiplies, it gets bigger, more numerous (like in 2x3=6, two multiplied by three equals six = two times three is six).

CAUSE= (coll.) Because.

POWER= Energy.

SUPPLY= If you supply power, you produce energy.

ELECTRIFYING= If something is electrifying, it’s really exciting and intense, like getting an electric shock but in a good sense.

YOU BETTER= (coll.) The grammatically correct phrase would be “you’d better” (= you had better + infinitive without TO) and it means the same as SHOULD, and is used to give advice. But the ‘D is disappearing and now most people simply say “you better”, even in the UK.
As an anecdote, it’s not uncommon to have little kids considering this a modal verb, so they produce shocking sentences like this one I heard:
- I better do it, bettern’t I?
Which is not so shocking if you consider this the perfect equivalent for this:
- I should do it, shouldn’t I?

SHAPE UP= (coll.) Get ready for this.
You can say this when you’re starting a fight, for example, to tell the other person that they have to get ready for what you’re going to do to them. In this context, Sandy is telling Danny that she’s going to eat him up, bones and all, so he better get ready for her attack.

MY HEART IS SET ON YOU= I love you.

TO MY HEART I MUST BE TRUE= I must be faithful to what I feel (so if I love you, I must show it and try to get you)

NOTHING LEFT FOR ME TO DO= You said everything, you did everything, so there’s nothing else I need to do, just say I love you too and go with you.

THE ONE= The person.
Notice that the conjunction THAT is optional, and in conversation it is often dropped, so they say “you’re the one THAT I want” and the chorus simply repeats “the one I want” (without THAT). And WANT may mean "want" or also "desire" (in a sexual way).

HONEY= /hʌnɪ/ This word means the sweet liquid produce you get from bees, but it is also used as a love word to address your girlfriend/boyfriend, husband/wife or whatever, or even to simply address a girl (which can sound irritating for some of them if there is no loving relationship of some kind).

YES, INDEED= Yes, of course. We use “indeed” to emphasize something, in this case, to emphasize YES.

FILLED WITH AFFECTION= Feeling a lot of love.

CONVEY= If you convey a feeling or a message, you express it.

BETTER TAKE= Short form for YOU'D BETTER TAKE = You should take.
BETTER TAKE MY DIRECTION= You should take (your affection) to me = You should put your love on me = You should love me.

FEEL YOUR WAY= If you feel your way, it means that you are using your touch (your hands) to know where you are moving because you can’t see. In this case, since Danny is not blind, what Sandy really says is “explore my body with your hands”, in two words: “touch me all over” (oops, that was more than two). We can also suppose FEEL is not about touching but about feeling (emotions), so it would mean: let your emotions guide you (to me).

GONNA= (coll.) Going to.

PROVE= If you prove something you demonstrate that something is true.

FATE= Destiny.

JUSTIFIED= If something is justified, it is reasonable, there is a logical explanation for it, it’s common sense.

SURE= /ʃɔ:/

DOWN DEEP INSIDE= If you think or feel something “deep inside”, that is the real thing you think or feel (sometimes as opposed to what you show to the others). Saying “DOWN deep inside” is emphasizing still more the intimacy of those feelings.
- She’s always very kind to me, but I know that deep inside, she hates me.