When you say "I happen to like it", it usually has the feeling that you like something despite everyone else's opinion. Sometimes there will be emphasis on the word "I" and the sentence might start with "Well." I think the dictionary meaning of "happen" in this instance is: to do, encounter, or attain something by or as if by chance. As I said before, it implies that you came up with this opinion on your own, not because of what others may think.

actually/unexpectedly/against the popular opinion/ no matter what you or anyone else says, I happen to like it :) Got to be old English writing style - it doth happen upon me this fine morning of October hehehehe I dunno :D

In this case "happen to" is used idiomatically to say that you disagree with the majority opinion. For instance: "Man, that movie was terrible." "Well _I_ happen to think it was great." / "New Jersey is the worst place in the world." "Well, I'm from Jersey and I happen to believe that it's a wonderful place."