Damn you Face, I’ll show you! I’m going to cut off Nose, and we’ll see who’s laughing then! Ha!
So this expression means to do something harmful to someone else, maybe to get revenge for something they’ve done. But in doing so, you’re also hurting yourself, probably more. I think it comes from the Middle Ages when people actually did this sort of self-mutilation.
Spite is a desire to hurt, annoy or offend. Related but with a slightly different meaning is the phrase in spite of, which I find difficult to explain to learners of English.
In spite of the rain, we enjoyed our time at the beach. = Up yours, Rain! You thought you could ruin our day, but we had a good time anyway!