The speaker is making a joke. Or at least is being mildly facetious. Instead of saying "I was introduced to three men, but the introductions sounded like a mumble to me and I couldn't understand them," he says "they were introduced to me as 'Mr. Mumble.'" Part of the point is that this is a familiar problem in parties, He expects us, the audience, to recognize a familiar situation. We understand that it is impossible to misinterpret a mumbled name as the actual word "mumble," to say nothing of then name "Mumble," but we get the point.

It means that the narrator didn't hear what the men's names were, and probably he didn't care what they were.