It's an insult inviting the listener to bite some unpleasant part of the speaker's body, usually the arse. Such an action is considered to be very unpleasant to the person performing the action, hence it is an insult. The fact that it might also be unpleasant for the person being bitten has somehow been overlooked. This phrase is similar to "Get bent," telling the listener to bend over in preparation to be... uh... violated in a rude way. Also a major insult!!!
I also remember that in the early days of this saying (in the 1990s), it was often spoken as an insulting 'suggestion' to the listener: "You can bite my a**."!!! Then it got shortened to "Bite me." By then, most Americans knew what the shortened saying meant.