A bristle is a stiff hair — the kind men shave off their face or the kind badgers have all over. Bristle also means to get angry. Tell an animal rights activist you use a badger's bristle shaving brush and you'll get the idea.
The emotional meaning of to bristle comes from the fact that most animal bristles used by man are so-called erectile hairs — the ones that stand up on the neck or along the back of animal when it's angry or surprised. A common word associated with bristle is hackle, another name for such erectile animal hairs. Thus the saying "to get one's hackles up," which is pretty much identical to bristling.