Because of the anatomy of the horse's teeth. There is very little overlap of the chewing surfaces, at rest, and there would be little contact of the chewing surfaces if the horse chewed up and down, like people do.
Horses have to chew side to side to get good occlusion or contact with upper and lower cheek teeth.
Horses can chew only one side on the mouth at a time. The horse opens the mouth slightly, moves the jaw to one side. As the mouth closes the chewing surfaces shear off or mince food. As the jaw returns to the resting position the chewing surfaces separate. Six upper and lower incisors or front teeth.