I upvoted MadWard, but as your comment indicates you are still confused, let me try to say it a different way.
"Happened to" in the sense you're using it means that something occurred by chance, or due to unexplained circumstances. Like if you say, "I happened to be alone that day", you mean that there was no clear, specific reason why you were alone. It's not that you planned to be alone. It's not that your family or friends all left you alone because they were angry at you. You just are alone, for no particular reason. Or, "John and I happened to be wearing the same color shirt." It's not that we planned this. I didn't call him up and ask him what color shirt he would be wearing, etc.
Note that when I say "unexplained circumstances", that doesn't necessarily mean totally random, but just unexplained in the present context. Like if someone said, "Hey, I can help you. I happen to be an auto mechanic", he probably doesn't mean that he became an auto mechanic by chance. Rather, he means that the circumstances that led him to become an auto mechanic are not relevant to the present discussion, and you probably don't know what they are, but the fact that he is an auto mechanic is relevant. Of course, presumably every event has some cause, so nothing is totally random anyway. We just don't know the cause. (And further discussion along that line could get very philosophical.)
"Ended up" means that something was the result of many steps. "I had an argument with my wife and she decided to spend the night with her mother. So I called my friend Bob but he was still mad at me for that prank I pulled on him last week. I went to see my old girlfriend Sally but she has a new boyfriend now. And so I ended up alone." It didn't just "happen", it was the result of a series of events.