Animal cells do not have cell walls because they do not need them. Cell walls, which are found in plant cells, maintain cell shape, almost as if each cell has its own exoskeleton. This rigidity allows plants to stand upright without the need for bones or musculature.
It is generally an advantage for plants to stand upright and grow as tall as possible. Plants need sunlight, so growing taller than neighboring plants make it more likely that the plant will survive and successfully reproduce. For animals, height may be an advantage sometimes as well, but most animals have skeletons and musculature. They do not need the rigid network that cell walls provide to stand upright.
Furthermore, most animals can move, and this capability is an enormous advantage when it comes to feeding, finding a mate and escaping from predators. If animals had cell walls in the same way plants do, they would be incapable of movement. Animals that do not move, such as coral and sea sponges, tend to grow in communal groups and develop structures that are very plant-like.
Animals and plants evolved differently, each according to different sets of needs. For animals, mobility was a greater advantage. Having cells without the rigidity of cell walls meant greater flexibility and motility.