What is an Anti-Corruption layer, and how is it used?

Question

I'm trying to figure out what the Anti-Corruption layer really means. I know that it's a way to transition/work around legacy code or bad APIs. What I don't understand is how it works and what makes it a clean separation from the undesirable layer.

I've done some searching, but I can't find any simple examples or explanations, so I'm looking for someone who understands it and can explain it with simple examples. An answer that would satisfy my question should be simple (not necessarily short), and provide understandable examples of implementation and use.

See this question, for my use case.