First, if you break water into oxygen and hydrogen, does that destroy water?
Second, does a fuel cell burn the oxygen and hydrogen to produce the energy that runs some kind of generator?
Yes, the water is broken down to its elements, oxygen, chemical symbol O, and hydrogen, chemical symbol H. In nature two molecules of water yield two molecules of hydrogen and one molecule of oxygen.
2H2O --> 2H2 + O2
In nature, you don't find single, free atoms of hydrogen and oxygen. However, they exist happily as what are called diatomic molecules. It's a chemistry observation, that's been known for ages. It's explained by their electrons behavior.
Second, does a fuel cell burn the oxygen and hydrogen to produce the energy that runs some kind of generator?
Yes, reforming the stuff back into two molecules of water and releasing the energy in those chemical bonds. Hydrogen likes to be in the oxidized state of water. Oxygen likes to be reduced state of water. In chemistry they are called reduction - oxidation, aka redox, reactions.