As you may notice, although he is American, he doesn't pronounce the R when it's not followed by a vowel (just like Afro-Americans or British do). That happens in some parts of America and it's kind of fashion to do it when singing (some singers pronounce the R when speaking but not when singing). Another striking feature (but not too unusual either) is the way he often pronounces the vowel in SHIP (/ ɪ /) in some unstressed syllables: he pronounces it like a SCHWA / ə /, so for instance he says "hesitate" /hezəteɪt/ or "complicate" /kɒmpləkeɪt/ or "vanity" /vænətɪ/.

YOU DONE ME= In this context, it probably means "you got me, you made me fall in love with you".

YOU BET I FELT IT= You can be sure I noticed.

CHILL= Cold.

HOT= If a person is hot they have a very attractive appearance. But if something's hot it can melt the ice, so here he uses HOT in both senses.

MELT= When you heat ice, it melts. So he tried to be cold as ice (use his head, not his heart), but she’s so attractive that he couldn’t stay cold ("you’re so hot that I melted": using "hot" here in both senses, attractive and high temperature).

I FELL RIGHT THROUGH THE CRACKS= He melted, so he became like water, so liquid that he fell to the ground and could even disappear into the cracks on the floor. The word RIGHT is emphasizing the idea of the word THROUGH.

BEFORE THE COOL DONE RUN OUT = This sentence is not good standard English. COOL is "nice, beautiful",  THE COOL DONE means "the cool things done", that is, the nice things still going on. The idea is plural, so the verb is plural (without -s). If something RUNS OUT it finishes because we used it all. So this sentence means: before all the nice things we did finish.

BESTEST (coll.)= The best of the best; better than the best. "I'll give my best" means that I'll try to be as good as possible.

I RECKON= I think.

HESITATE= to doubt, to be uncertain.

CANNOT= Notice that this is the only verb that can’t be written "can not", but all in one single word, and it’s pronounced /kænət/.

IT CANNOT WAIT= We can't postpone this (we must jump into a relationship right now).

OPEN UP YOUR MIND= When you open up your mind you get more flexible: you can accept different ideas, imagine different possibilities, be ready for new things.

SEE LIKE ME= See the situation the same way as I do (with an open mind).

OPEN UP YOUR PLANS= Make your plans flexible, be ready to accept a different future and change your plans.

DAMN= A bad word, but soft, not really offensive. It’s often used as an exclamation when you don’t like something (e.g. "oh damn! the television is broken") and also, like in this song, used to emphasize something ("damn! you’re free")

WE’RE JUST ONE BIG FAMILY= When you’re in love you feel you are in harmony with the universe, you feel connected with everyone and everything, you feel like all humankind is like a big family, and you love them all.

IT’S OUR RIGHT TO BE LOVED= We deserve to be loved; it is just and fair that we are loved. The word "God-forsaken" (another rude word) is used here just to give emphasis to the word "right".

FATE= Your fate is your inevitable future, what is predestined for you; your destiny.

SCOOCH OVER CLOSER (coll.)= Sit closer to me; move nearer. The particle ON gives the idea of continuity, so "scooch on over closer" means "keep approaching me".

NIBBLE= To bite at gently and repeatedly.

WAY TOO LONG= Too too long; an excessive amount of time. We often use the word WAY to emphasize TOO.

CHECK= If you check something, you look at it carefully to get some information about it or to see if you like it or not.

TONGUE= Pronounced /tʌŋ/.

BENDING OVER BACKWARDS= This is a very complicated body position, here's a picture to show you what it looks like: picture. In English we can use this as an expression meaning that you are willing to do every possible effort for something, usually to help another person (e.g. "he's always bending over backwards to help me"). This would be a strange comical position to look at your tongue in the mirror, but he just means he was doing everything he could to thoroughly check his tongue carefully. What for? I read something about AIDS that might explain this:
One of the typical symptoms that may lead to AIDS diagnoses is the appearance of a white area in some part of your tongue, often at the back or below, so some people afraid of AIDS would regularly check their tongues in the mirror just in case. In that case, this song doesn't say "forget about AIDS, it's worth the risk", but rather "don't be so scared about death and live life fully while you have it". By the way, Jason Mraz is not gay, though there's quite a lot of buzz about it. He is (or just was) in a relationship with a girl but yes, he likes to joke about that and says he's "psychologically bisexual" but not physically so. This song is not about loving a man, or being gay or living with AIDS, it's about living the present carelessly and fully. The reference about checking your tongue is just a reference who people who are physically or mentally hypochondriac (too worried about illnesses).

MY BREATH= The air that goes out of your mouth when you breathe. "To breathe" /bri:ð/; "breath"/breθ/.

FOGGED UP THE GLASS= If something fogs up the glass, the glass is covered with a coat of condensed water, like mist, and you can’t see through it.

LAUGHED= /læft/ (standard BrE: /lɑ:ft/)

AIN’T (coll.)= The negative form of the present tense of "to be" or "have":
- I ain’t got no money = I haven’t got any money
- He ain’t French = he’s not French
- THERE AIN’T NO BETTER REASON = There isn’t a reason better than this one.

TO RID= If you rid or get rid of something, you throw it away.

GO WITH THE SEASONS= To go with the season is to match the mood of the season, so to go with the seasons (in plural) probably means to change, to adapt, to be flexible, to stop trying to control everything and just relax and let yourself go.

IT’S WHAT WE AIM TO DO= It’s our objective, what we plan to do.

VIRTUE= Pronounced /vɜ:*tʃu:/.

OUR NAME IS OUR VIRTUE= He possibly means that what we are (our identity) is the important thing, not all our vanities (possessions, adornments, clothes, poses, etc.).

‘CAUSE (coll.)= Because.