Why did shi huangdi support legalism?

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.