(1) I am to explain the position.
This is about intent: I will, I'm going to. It may also denote duty: *it is my task, to explain..." but not necessarily. As others mentioned, it's old-fashioned form, and rather less formal (formal versions wouldn't skip the verb: "I am supposed to..." or "I was tasked with explaining...").
(2) I have got to explain the position.
You won't see this sentence in the wild. "got" is thrown in after "have" in informal speech, but then, in informal speech "I have" is always contracted to "I've". What you will see is:
(2) I've got to explain the position.
This is a fairly common form: "I have no other choice, but to explain the position". It'd about compulsion, duty, need - it's not just that the speaker will explain; the speaker can't refuse explaining (or not doing so would be very undesirable).
(3) I have to explain the position.
This is equivalent to the above but acceptable in formal contexts. While you wouldn't write "We've got to test..." in a scientific paper, writing "We have to test" is perfectly fine. This is also normal in common (non-formal) speech, possibly carrying slightly more emphasis than (2). Note, in this case "I have to" is almost never contracted to "I've to".