When a horse pulls on a cart, the cart pulls on the horse with an equal but opposite force. How is the horse able to pull the cart?

The horse pushes on the Earth. You need to think about that for a while, the horse and cart are not the only things in the system.

Horse pushes the earth, true.

But that alone will not move the horse forward.

What moves it forward is the force of friction between the horse and the earth.

Without friction, the horse will keep slipping without moving forward.