Question

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

My Answer:
The force of the horse pushing on the ground overcomes the inertia and frictional forces acting on the cart.

Is this correct? Should I add more?

Answer 1

You're correct, but I disagree at particular part of your sentence.
"The force of the horse pushing on the ground overcomes the inertia and frictional forces acting on the cart."
The force horse exerts on the ground doesn't overcome cart's inertia. It only "helps" the horse not to slip while horse used another force to overcome that inertia :)

Answer 2

Answer is simply that these forces you are mentioning are exerted on DIFFERENT systems, so when you are studying the motion of the cart, the force it exerts on the horse has not to be taken into consideration.