The horse pulls the cart with the same force as the cart pulls the horse in the opposite direction. So how does the cart move?
The horse pulls the cart with the same force as the cart pulls the horse in the opposite direction. But the horse pulls the cart diagonally by stepping on the ground. Consequently, a force acts diagonally on the ground.
The ground also exerts a reaction force along the foot. This reaction force can be divided into two perpendicular components. If the magnitude of the component acting horizontally along the ground is greater than the reverse reaction force acting on the horse by the vehicle, the horse will move forward.
On the other hand, the ground friction force prevents the vehicle from moving. So if the force exerted by the horse on the car is greater than this frictional force, the car will start moving forward.