"I have got to go" is slightly more formal than "I have to go", but not by a huge amount. Also, "I have got to go" can mean than you have no choice, and emphasizes the fact you have no choice. Example; Person A: Did you tell her that you don't want to go to the party? Person B: I did, but I don't have any other option, I have got to go or I may lose the opportunity. Person A: Where are you going? Person B: I have to go to the shops, did you need something? It's a very, very minor difference but sometimes saying "I have got to go" is a way to emphasize the lack of choice, where as "I have to go" can often relate to time, not decision. Hope it helps.

I have got to = I must = I have to

The thing which is going to happen (or expected) is expressed in "to go", not in the present perfect construction. "I have to..." --> "I've got to..."(notice the contraction!) --> "I got to..." So in this context, "have" has been sucked into the pronoun, but we still need some kind of working verb for the sentence. "Got" seems to work fine. This comes from spoken English, not written English, by the way.