(1) I am to explain the position.
(2) I have got to explain the position.
(3) I have to explain the position.
Please tell me when the meaning of above sentences is one and the same. Also, why and under what circumstances we have to use the above sentences.
They are somewhat different.
If you say, "I am to explain the position", I'd think that you are going to explain me the position.
If you say, "I have got to explain the position", I'd think that you must do that.
If you say, "I have to explain the position", I'd think the same as the second one [Oxford defines it the same as have got to].
(1) I am to explain the position.
(2) I have got to explain the position.
(3) I have to explain the position.
The sentence (1) is a little old-fashioned, and means roughly that you are going to explain the position, according to some plan. It's not very different from "I will explain the position," in fact is probably not at all different, but it does suggest that there is some plan out there assigning you to explain the position.
The sentences (2) and (3) are synonymous, and mean that you are required to explain the position.
(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".
I disagree with the hair splitting that others have engaged in to answer your question. The answer is simply this: the examples you posted are equivalent and interchangeable.
[although this is also an answer, i'd leave this as a comment if i could.]
