If action and reaction are equal and opposite, how does anything move?

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Newton's third law is correct: when the horse pulls the cart forward, the cart pulls the horse backward with an equal force. But these two forces act on different bodies, one on the cart and one on the horse. Forces cancel only when they act on the same body, so a third-law pair can never cancel.

To decide whether something moves, list only the forces acting on it. The cart feels the horse's forward pull and a backward resistance from the ground (friction at the wheels). If the pull is larger, the cart accelerates forward.

The horse feels the cart's backward pull and a forward friction force from the ground. The horse pushes backward on the ground with its hooves, and the ground pushes the horse forward (that is another third-law pair). If the ground's forward push on the horse is larger than the cart's backward pull, the horse accelerates forward too.

So for the horse and cart taken together, motion is decided by the external forces from the ground. On frictionless ice the hooves would just slip and the pair could not start moving. The rule to remember: third-law pairs are equal, opposite, of the same type and on different objects, so apply F = ma to one object at a time.