When One Person Tells Another Person To Bite Me, What Does It Imply?

It is a term meant to be an insult, or to tell someone off without the foul language. Something derogatory. Hope this helps.

It does not mean that they want to literally be biten. It means that are insulting you. They are telling you this because you said something to make them mad or upset. It is an offensive way of insulting someone. It does not mean to literally bite someone. It really means "F*&^ you" (excuse my french), but using nicer, cleaner words. You should turn and walk away from someone who tells you this.

Well it ain't an invitation to lunch. It in recent years has taken the place of another phrase that one would invite  another to smootch. After a while people found that phrase was not insulting but a pleasure. Lol

"Bite me" is a polite way of saying "blow me".