When a person says this about themselves, it is a light-hearted expression of being pleased with themselves for.say, having risen to some challenge and really succeeded; or having handled/dealt with some situation and really come out on top. It could be that they are generally regarded as a reserved person; and then with a couple of drinks in him, he does a karaoke number at a party which everyone loves and thinks is tremendous. Someone tells him he was great, they never knew he was such a good singer, to which, rather than puffing himself up and bragging in some way, as if lightly dismisses the praise with a smile and says, "Oh, I have my moments." I guess you could say, it's moments/flashes of a kind of 'brilliance" or "greatness" (but on a small scale).
However, when said about, say, a film, it is not so wonderful. A film may have been generally hum-drum or boring, and the best you can say that there were a few comic moments, or a few interesting ideas in the film - 'It had it's moments." We have another expression: damned my faint praise. So - if you asked someone what they thought about a movie, and they said, it has its moments, you gather that the movie is pretty poor and he is not recommending it.
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