Help me understand why water atoms will break ingto...?

What happens is that the battery gives one electrode extra electrons, and takes electrons from the other electrode. When the water (H2O) breaks into OH- and H+, the H+ combines with an extra electron and forms hydrogen gas. Meanwhile, the salt (NaCl) breaks into Na+ and Cl-. The Cl- gives it's extra electron to the positive electrode and forms chlorine gas. The secondary reaction that they mention on that page is when the Na+ and the OH- combine to form NaOH.