If Action Is Always Equal To The Reaction, How A Horse Can Pull A Cart?

The horse has to exert more effort to pull a cart than it would if it were walking unhindered.  So the backward force the horses hooves have to exert on the ground result in an equal force moving forward.  The heavier the cart, the more force has to be exerted by the horse.  That is why, when going up a hill, considerate people will get out of a cart and walk, rather than make the horse pull the weight PLUS fight gravity.

To correct you both horses don't pull anything, They push loads, carts etc through their shoulders. That's why if you watch them they lean forward into the collar (the part around the neck just forward of the shoulders)when moving an object.