It is the same as "must" We must go outside now - I must get my wallet back.

Have to(must do something) = Got to. 'Got' unfortunately seems to have replaced 'have' and often people say 'have got' which is unecessary, it's basically using two words together that mean the same thing. In formal English we would say 'I have a dog' but many people these days would say 'I have got a dog'. Many of the c**bridge esl textbooks teach this as correct grammar although a large number of esl teachers from the Commonwealth disagree with it's use. Even if we use contractions we don't have to use 'got' we could say . I've a dog'. The past tense of got is of course 'have' which reinforces the point that 'got' isn't necessary when you are using 'have'. But it all makes sense when we watch action films! Gotta go! Bye!

"Got to" is an idiom meaning "have to" or "need to." In fast conversation, it sounds something like this: "Gotta go now." (meaning "I have to leave immediately.") We run the "got" and "to" together.

It's an add-on that signifies an absolute necessity or need to do something or obtain something. It doesn't just mean to get something back or in return, it also can be used with just about any verb. "I have got to pass my math test." "I got to eat. I got to drink. I got to dance. etc. It's more a slang term, so in professional or formal situations i would refrain from using it. Also answering you're question as to why it's written in a past-tense form, it doesn't have a reason really, it's kind of an exception to the rules as are many things in the English language.

I got to.. could mean "I should..", "I must.." or "I need to.." For example, "I got to go. I am late." You could change this to "I should go", "I must go" or "I need to go".. "got to" is usually used in informal English or in a normal conversation..