A UK and Australian idiom. It means 'more than is normal, required or expected'.

It is most often used to refer to cost, as in "he payed over the odds", which could mean either that he payed too much money (was cheated) or that he payed more than he was expected to, out of the kindness of his heart.

Being a stupid American, I've never heard this expression used, although I've come across it in written works from time to time (not in American writings). I haven't been able to find any explanation of where it came from, but it probably comes from the more common "triumphing over the odds."