I LIKE TO MOVE IT= The verb LIKE (as all verbs of likes and dislike) is followed by –ING (e.g. I like swimming, I love dancing) except when they go with "would" (I’d like to dance). But we can also use these verbs with infinitive with TO, with a change of meaning:
- I like dancing (= when I dance, I enjoy dancing, it’s great)
- I like to do exercise (= I think it’s the right thing to do. I think it’s a good thing. Maybe I enjoy it or maybe I don’t, but I think it’s good).
But sometimes, this difference disappear. In this case, they probably used the construction with TO because it fits better with the rhythm of the music, but the meaning here is the same as "I like moving it". The IT here refers to your bottom (the part of your body where you sit down), so "I like to move it" means here "I like dancing".

HERE WE COME = There we go = We’re coming, get ready!.= They say "here we come, baby", so they’re calling New York "baby", just like it was a person, not a place.

SOUTH OF MANHATTAN= This phrase is used as a name, just the same as we could say "London" or "The Isle of Whight", so there is no article. If they were talking about Manhattan and they want to specify which part of Manhattan, they would say "if you come to live in the south of Manhattan..." ("the south", with an article).

FEEL FREE TO CALL FIRST= This sentence is ironic, the usual sentence would be: "feel free to come to visit", which means I extend an open invitation for a visit, you don’t have to worry about asking first, just come and visit any time. Here, they use the same sentence but they change it into "call first", so it means: "don’t worry, you are free to... try to visit me, but phone me and I’ll tell you if I invite you or not", so the "feel free" part is only mockery. The sentence "if you ever come to liven in South of Manhatttan, feel free to call first" really means: I don’t like you, so please, come to visit if you want, but I won’t let you come to my house. Then he emphasises this idea without irony, in plain English: "seriously though, call" (though= nevertheless).

FREAKS= A freak, or a freak person, is someone who is very strange in an unnatural way.

BUCKLE UP, BOYS!= A belt is a strap of material you fasten around your waist to keep your trousers/skirt up. The BUCKLE is the part of the belt you use to fasten both ends together (se picture here: buckle). Here, they refer to the seat belt, the one you use to secure yourself to the seat in a car or plane, to protect you in case of accident. The verb TO BUCKLE means to fasten your belt or seat-belt. TO BUCKLE UP is the same as TO BUCKLE, especially with seat belts.

HAIRY= Scary, frightening. When you are very afraid, your hair stands on end.

LAUNCH= Planes take off, but rockets launch (or are launched). This plane is not going to take off by itself, it’s going to be propelled into the air, so they use the verb "launch", like for rockets.

THE GOOD NEWS IS...= Notice that NEWS is a singular noun ending in S (actually, it’s an uncountable noun), so the verb is in the singular form. We can say "I’ve got news" or "I’ve got some news". If we want to make it countable we can say: one piece of news, two pieces of news.

CRASH LANDING= The verb CRASH means to smash, to bump into a place and suffer some kind of destruction. But here it is used as an adverb, because they put it before the verb "landing": "we are landing – we are crash landing, so "crash" specifies what kind of landing they’re going to have: crashing.

DEPLOY= To organize (troops or resources) into a position ready for immediate and effective action. Here, it refers to the parachute.