Others might interpret this differently, but I think that she's still not over her last boyfriend. Have you ever heard 'missing someone was like losing an arm'? I think that even though Matt's character really loves her she can't let go of her last love, and when Matt's character finally gets fed up and 'chucks' the other guy he then feels bad. Then trying to make amends he makes a 'sacrifice' to give her of a piece of himself.

I suppose it sort of plays on the idea that you can never have a relationship with someone and not give up a small piece of yourself or possibly leave a piece of yourself behind when its all over.