"Found out to his cost" means that a person finds something bad out, and that that bad thing affects them (and always had been, without their knowing so). It may be a financial "cost", but is not necessarily so, as it can be a metaphor. For example: the British retailer may "find out" that the product that he buys expensively from France, in order to sell it in his shop, is actually made by Spain much more cheaply. This causes his business to fail, because other shops buy from Spain. So he will 'find out to his cost' that buying from France is a bad idea.

I think this one is British-only. I've never heard it in the states.