A chicken is a ground-dwelling bird, not a flying one. It uses its leg muscles for unhurried, long-duration movements such as roaming around the barnyard searching for insects and other food. In contrast, a chicken hardly uses its wings except for balance. When it flaps them energetically, it's usually to make a quick escape from a threat. Because the muscle requirements of the chicken leg and breast are different, the two sets of muscles evolve differently. The legs consist predominantly of slow-contraction muscle fibers, while the breast is composed chiefly of fast-contraction muscle fibers to help flap the wings.
The slow-contraction muscle fiber is for the long-duration jobs and the fast-contraction muscle fiber for the quick-energy spurts. The fuel for the slow-contraction muscle fiber is fat and requires oxygen, which is stored in the iron-rich, red-pigmented myoglobin. Consequently, the more slow-contraction fibers in a muscle, the redder the muscle will likely be. Fast-contraction muscle fibers don't require oxygen that much — and therefore myoglobin — because they use glycogen (a carbohydrate) for fuel. In the absence of myoglobin, the muscles are "white".
Actually, both the chicken leg and breast contain a combination of fast- and slow-contraction muscle fibers. The leg is "dark meat" because the slow-contraction muscle fibers predominate. The opposite is true for the breast.