For instance, if you were talking with a small group of friends and the conversation turned to how much they dislike the city you all live in - but you like it - you might say, “well, I happen to like it here.”

I happen to like it here suggests that 'here' is a place that is not usually liked, or that it is somehow surprising that you like it here. I like it here simply states a fact with no other suggestion.

Broadly & coldly speaking, there's no difference. It's mainly a tone & style thing. The exact nuance of meaning also depends which word you emphasise. If we want to magnify the expression even more, we could say: As a matter of fact, I just so happen to actually like it here to be honest.