More formally, transubstatiation can be explained by distinguishing between form and substance. Transubstantiation is simply a process where the substance changes, but not the form. However, it must be stressed that the terms form and substance are not used in the modern sense of shape and material. These terms are used in the context of Aritstoelian/Scholastic metphysics.
Contrary to bigeldac's comment, transsubstantiation is not caused by belief. That is, even if I really believe that can of Coke is really Pepsi, that can is no less a can of Coke than it was before. It is the Catholic's belief, however, that by the grace of God, when the priest utters the words of the sacrament, the substance of the bread and wine become body and blood of Christ---whether one chooses to believe it or not. That is why it's considered a miracle.
There have been cases where reportedly, the (form of the) bread actually changed to real flesh, and the wine, blood. Although I am a Catholic, somehow the idea grosses me out.