Legalism became the official philosophy of the Qin Dynasty when the first emperor of China, Shi Huangti, rose to power and banned all other philosophies as a corrupting influence.
If you study the history of the Chinese emperors, you'll notice that this is a pretty common thing amongst them all. (ex: the mandate of heaven which outlined things rulers needed to do to remain "virtuous") Basically, every new dynasty came up with a new set of rules outlining exactly how the previous dynasty was evil.
Your simplified answer: He supported his policies of legalism because he thought all other philosophies to be corrupt.