Question

Why did the political chaos of the Period of the Warring States give rise to philosophies such as Confucianism, Legalism, and Daoism?

Answer 1

Daoism grew out of native Chinese shamanism and folk medicine. As such it was already in existence during the period of the Warring States. Lao-Dzu was a genius who encapsulated Daoism into a book about 5,000 years long.
Buddhism didn't reach China until about the 8th century, through the bi or trilingual traveler Fu Hsien (I think that's his name).
As for legalism, this was the perfect prop for any state. The law is nearly sacred; hence breaking the law is almost treasonous. Actually this reflects ancient Asiatic and Mid Eastern beliefs; for each religion its laws were given by God.