It's an American idiom, it means "I don't care what you think." It could also translate to "go to hell" or "go away and is used to express annoyance or dislike. for example: "I told one of my students she needed to bring a pen with her to class and she said "bite me!" I couldn't believe how rude she was! She's talking to the headmaster right now. I hope there's going to be some serious consequences."

It's very rude. Don't try to use it. It carries an obscene meaning. It's used when someone is angry and is denying a request. It is an angry way of saying "I'm not going to do it, you have no right to ask, and you can't make me." It's an insult because the person says it is angrily rejecting your request without even thinking about it. If someone says it to you, you'll be able to tell from their tone of voice that they're angry.