Cookie Monster speaks broken English, or rather, a personal variety of English which reminds us of pidgin, creole or even Black American English dialect, that’s why he uses simplified grammar and modified pronunciation. The most outstanding feature is the use of ME for the subject instead of the standard I, but that is also common in the English varieties already mentioned.

ME= I
This is common in pidgin, creole, Black AmE and other varieties of English used by (or originally by) non native speakers.
- Me don’t like da movie = I don’t like the film

ME= My
This is in fact an old English form of the possessive adjective (before it was diphthongized) still surviving in many English dialect, especially in the north and Scottish English. From Scottish foremen, it came into black AmE.
- I miss me dad and me mum = I miss my father and my mother
- Das me car = That’s my car

IT IS A CHOCOLATE CHIP KIND= [My wish] is a chocolate chip (that kind of wish).
A CHOCOLATE CHIP is a kind of biscuit/cookie which has chips of chocolate inside (see picture). The use is different in BrE and in AmE:
AmE Cookie = BrE Bicsuit /bɪskɪt/
AmE Chocolate chip = BrE Cookie

SNICKERDOODLE= A special kind of biscuit with cinnamon, usually made for Christmas (see picture)

ME TRADE ME SOUL FOR A BITE= (a colloquial exaggeration) If you trade (exchange) your soul for something, you want it very very much (you could give your soul to the devil in exchange for that, like Fausto). A bite is a piece of something to eat, a morsel.

ME SPELL IT OUT BLACK AND WHITE= If you spell something out black and white (like black letters on a white piece of paper), you are saying it very clearly, hiding nothing.

YOU, COOKIE-SHOWING= You are showing biscuits.

ME, HUNGER GROWING= I am growing hungry (I’m getting more and more hungry)

LET’S GET SKIM MILK FLOWING= Let’s drink skim milk (if a liquid flows, it moves, in this case down into your stomach). SKIM MILK (skimmed milk in BrE and Canada) is a kind of milk without its cream (fat). The basic kinds of milk are called differently in BrE and AmE, these are the equivalences:
BrE whole milk (4% fat) = AmE whole milk (3.5% fat) = milk with all its natural cream (more or less)
BrE semi skimmed milk (1.7% fat) = AmE Reduced Fat milk (2%) // Low-fat milk or Light milk (1% )
BrE skimmed milk (0.2%) = AmE skim milk, non-fat milk, fat-free milk (0%)

YOU GOT COOKIE= (coll.) You’ve got biscuits.
(here, “cookie” is used as an uncountable noun, a stuff, so it has no plural)

SHARE IT MAYBE= Would you like to share your biscuits with me? (maybe you would?)

CUPPY-CAKE= A dessert which looks like a muffin with a lot of frosting on top (see picture).

FROSTING= A kind of solid sugar cream used to decorate or cover the top of many cakes, sweets and desserts.

IT NO PHASE ME= (coll., usually used in the negative) It doesn’t disturb me or bother me = pie and ice-cream are ok, I like them too.
Using NO to put a verb into the negative (instead of NOT or DO NOT) is also common in pidgin languages

YOU TOOK YOU TIME WITH THE BITE= You took a long time to give me a bite, or maybe I was waiting for a long time and you never gave me the bite.

FREAK OUT= Panic.

GIRL SCOUT= (masculine: boyscout) A girl belonging to the boyscouts association, which organizes camping activities etc for children. (to scout = to explore)
Girl scouts are famous for selling biscuits to raise money for their activities.

GRUMBLE= To complain in a low indistinct tone, like mumbling.

GROUSE= To complain in a low indistinct tone, like mumbling.

IT’S TAKING TOLL ON ME HOUSE= If something takes a toll, it’s not free (toll = tax): usually used in a figurative way meaning “to have a negative impact”. If something takes a toll on your house, you have to mortgage your house to get all the money you need, so basically, this expression is just an exaggerated way of saying that it had a very negative impact on me (in this case because waiting for the biscuit is driving him crazy).

ME, GOING OFF ME ROCKER= I’m going mad (AmE crazy)

BETTY CROCKER= A famous brand of chocolate chips (BrE cookies)

ME MISSED YOU SO BAD= I missed you very very much.
We use “bad” with some positive feelings to express that the feeling is very strong:
- Oh baby, I love you so bad (I love you very very much)
Maybe the original idea is that the feeling is so strong that it hurts (that’s why we began saying “bad”)