What makes the hair of some animals like dogs and cats to stand upright?

Answer 1

Each hair shaft grows from a follicle. Each follicle has a muscle associated with it called erector pili (Latin for "hair stander-upper"). When the muscle contracts, it makes the hair stand up. The muscle contracts when stimulated by cold, fear, or aggression.

Answer 2

The same way any mammal does (and that humans get "goose bumps"), there are very tiny muscles attached to each follicle that can tilt the hairs upward (to make them stand on end) or downward (to make them lie flat).

These muscles are controlled automatically to regulate body temperature and in some animals also voluntarily to allow them to change their appearance.