Why is the cheetah becoming extinct?

Well it's not completely known why the American Cheetah went extinct. They likely died off with a large number of other North American animals (American Lions, Horses, Mammoths etc.) at the end of the last ice age, or in the Younger Drias event that followed shortly thereafter.

Some ideas as to the cause of the mass extinction are the abrupt climate change, human pressure, or a possible meteoric/comet impact or pandemic.

Improvement:

The American Cheetah is actually not a Cheetah at all and has no relation to the Cheetah we know now. It was closely related to the Mountain Lion.