The horse moves because the force he exerts with his hooves is greater than the force of the wagon pulling him back.

What pushes the horse forward is the ground. The horse pushes backward on the ground, so the ground pushes forward with an equal force. If the horse can push back against the ground with a force greater than the carts resisting force, then the horse will accelerate.

Acceleration will occur if one force pair (push of ground/push of horse) is greater than another force pair (friction/pull of cart). 

The horse pulls it forward, and there is a backward force from the ground: friction. If the horse pull exceeds the friction of the cart, it will accelerate.