That bread has a lot of meaning.
First, we do that in remembrance of Him.
Second, I think we are to be bread. We are to be bread to others, just as he is bread to us.
Third, we are broken. We break off pieces of that bread to remind ourselves that we are broken, fallen and Christ loves us anyway.
In many ways, the Last Supper never ended. The church, His Love Body, his people are being broken and offered again and again.