Getting goose bumps is a reflex, which is an action your body has automatically without you even thinking about it. This particular reflex is known as the pilomotor reflex. Humans aren't the only mammals that have this reaction.
Function of Goosebumps
As for animals, the pilomotor reflex in them often has the effect of making an animal look bigger. This might help to scare away potential enemies that may have caused the fear reaction in the first place.
Goosebumps experienced in times of cold is the body's way of preserving its own heat by causing the hairs on the skin to stand up, thus reducing heat loss.
Why do the hairs stand up
When you get cold or experience a strong emotion, your brain sends signals to your muscles that make them tense up. When the muscles in your skin that are attached to hairs do this, they make the hairs stand up and pull your skin up just a bit, creating goose bumps.
Is there some primal benefit that no longer benefits us?
The formation of goose bumps in humans under stress is considered by some to be a vestigial reflex; some [who?] believe its function in human ancestors was to raise the body's hair, making the ancestor appear larger to scare off predators.
Do other animals experience this?
Yes, animals too experience goosebumps. For example, when porcupines are threatened, their quills raise in a reflex action. Similarly, you may have seen a cat's or a dog's hairs stand on end when they sense danger or feel afraid.