Because in order for animals to breath they must inhale oxygen, we get about 20% from all the air we inhale here on earth. Although there is some trace amounts of oxygen floating around in space and on the moon, by in large it is almost completely empty, and thus no one would be able to get the oxygen they need to survive.
Because there is no atmosphere on the moon, there is no oxygen. We breathe out carbon dioxide, right? Don't trees take in that to produce oxygen? There aren't any trees on the moon, and like I said, no atmosphere. The process wouldn't work. Therefore, no. We cannot breathe on the moon.