To "break bread" means to have a meal with someone, breaking off pieces of your loaf of bread to ensure that everyone is fed.
Jesus would eat with his disciples and give out pieces, and since bread back then was harder than modern bread, it was necessary to break it rather than tear it.
Fellowship is where His disciples once again gather for a family meal together. Breaking the bread together literally means becoming one body in Christ, an unbreakable family bond, not just with one another, but also with God.
We love sharing a meal together and fellowshipping in this way.