Horse pushes the ground in backward direction. In reaction to this, the horse moves forward and cart moves along with horse in forward direction.

In this case when horse tries to pull the cart in forward direction, cart pulls the horse in backward direction, but to make the cart move, the horse bends forwards and pushes the ground with its feet. When the forward reaction to the backward push of the horse is greater than the opposing frictional forces to the wheels, the cart moves