Two potential reasons I can think of:
1) They don't remember exactly what they said, so best to apologise in case they said something offensive.
2) It's not considered polite to talk about some very personal topics with someone you don't know, and people here can often find that an uncomfortable situation. They're apologising in case they made you uncomfortable with the level of openness they had while drunk. IME Americans tend to be far more open about personal topics even to complete strangers, whereas Brits are more reserved about sharing certain things.
I think if this happens again, just say something like, 'I wasn't offended at all, I quite enjoyed our chat.' Unless they really did say something offensive or something very personal that they wouldn't want repeated, I imagine it's more that they're trying to be polite and wanting to avoid YOUR discomfort rather than wanting to pretend like the conversation never happened for their own sake.