If action is always equal to the reaction,

Solution:
When a horse pulls a cart with a certain force, then the cart pulls the horse with the same force and the system of horse and cart should not move.
But, if we analyse the problem carefully, then horse not only applies force on the cart, but also pushes the earth backward with his feet. The earth reacts back to this action of the horse and pushes it in the forward direction. 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 move in the forward direction.
This unbalanced force applied by the earth makes the system of horse and cart move in the forward direction.