First of all, It's not a nice thing to say (yeah, it's rude) and it means something like "kiss my -pretty- *** " "get lost" "leave me alone" ... normally used when you DON'T CARE about what the other person (or the entire world) is saying to you. :-)

Wikipedia define: Bite me is typically used as an idiomatic expression of discontent or annoyance with another party

It's similar to "Leave me alone!" or "Go away!" . it's not considered especially vulgar, but usually inappropriate in formal settings. Politely you could say: "Please leave me alone." ciao