First, when the horse pulls o the cart, the cart exerts an equal but opposite reaction on the horse, the action and reaction. If this was the only force in action then the force and cart would indeed remain stationary.

However, there is another force between the force and the ground. The horse’s hooves press down on the ground and the ground pushes back on the horse.

If the reaction force of the ground is greater than the reaction force of the cart on the horse, then the horse will move forward. The cart will move forward when the force exerted on it by the horse is greater than the frictional force between the cart and the ground.