Better you than me is a way of saying it was a good thing it happened to the person you're talking too (you) rather than to yourself. It's usually when whatever happened is bad, but if it's say your friend, you may use it for good things too. Another way to say it is "It's better it happened to you than to me" but that sounds really awkward and wouldn't actually be used, I'm just using it as a way to show what it means. "Good thing it was you instead of me," could also be a valid synonym.

It's just another way to say "rather you than me". "Rather you than me" is more correct

It's often used to mean, "It's better that something bad or unpleasant happens or will happen to you, rather than to me." Example: John: "My wife makes me take her shopping every weekend for hours and hours. I can't stand it." Paul (glad that his own wife doesn't do that): "Better you than me." In that particular circumstance Paul could say, "Too bad about you, but I'm glad my wife doesn't make me do that." However, "Better you than me" conveys less sympathy and less concern for the other person.