A horse pulls forward on a carriage with a given force. By Newton's Third Law, the carriage must be pulling on the horse backward with an equal and opposite force. Given this, what explains why the horse and carriage can move forward?

Because total force applied by horse is greater than the force applied by carriage on horse.

Explanation:

Here, the force applied by horse to move the carriage is greater than the force applied by carriage in opposite direction. He exerts greater force with his hooves thus accelerating the cart.

The horse is applying some force to the ground, meanwhile ground is also pushing the horse with same amount of force but as the force crosses the threshold of carriage's resisting force, it started to move.

In the explanation

Explanation:

when we take the case of horse pulling the cart.

When the horse applies the force on the cart suppose it to be 'R',.

then the same reaction force of 'R' cart will apply the horse.

But as the horse use traction from the ground and the wheel of the cart does not have that i. e. it leads to the movement of cart.