Why do some people ride with fringes on the bridle or breastplate?

It's a traditional means of fly prevention. (Zebra stripes are also supposed to deter flies). The fringes deter flies from landing on the horse by the way they move around and are sometimes also coated with a chemical repellent.

Fringes, particularly on breastplates, are also used for pure aesthetics. That is to say, some people think they look good. You'll often see fringed breastplates at rodeos, in a variety of colors and sometimes with beading.