It is true that when a horse pulls a cart with a certain force, then the cart also pulls the horse with the same force and the system of horse and cart should not move.

However, if we analyse the problem carefully, then horse not only applies force on the cart, but also pushes the earth backward with his feet. In this action of the horse, the earth reacts back and pushes it in the forward direction. Now, the force applied by the horse on the earth is insufficient to move the earth, but the force applied by the earth is sufficient to make the horse to move in the forward direction.

It is this unbalanced force applied by the earth, which makes the system of horse and cart to move in the forward direction.