There are many reasons for that. Some of these are:
It has been suggested that the stripes serve as visual cues and identification. With each striping pattern unique to each individual, Zebras can recognize one another by their Stripes.
Another aspect given by the researchers is that that the Stripes serve a Camouflage Mechanism, according to which the vertical Striping helps the Zebra hide in grass. Though grass is neither white nor black, it is supposed to be effective against the Zebra's main predator, the Lion, which is colour blind. Theoretically a Zebra standing still in tall grass may not be noticed at all by a Lion. Zebras are herd animals and the Stripes to some extent help in confusing the predators - a number of Zebras standing or moving close together may appear as one large animal, making it more difficult for the Lion to pick out any single Zebra to attack. A herd of Zebras scattering to avoid a predator will also represent to that predator a confused mass of vertical Stripes travelling in multiple directions making it difficult for the predator to track an individual visually as it separates from its herd mates.
A recent theory, supported by experiment, concludes that the Stripes are also an effective means of confusing the visual system of the blood-sucking Tsetse Fly.
Some theories suggest that the Stripes coincide with fat patterning beneath the skin, serving as a thermoregulatory mechanism for the Zebra and that wounds sustained disrupt the striping pattern to clearly indicate the fitness of the animal to potential mates.