Newton's third law says the forces of an action-reaction pair are equal and opposite. Why doesn't every pair of forces cancel, leaving nothing able to accelerate?

A. Because the two forces act on different objects.

The second law is applied to one object at a time, and the reaction force acts on the other object, so it never appears on the first one's free-body diagram and cannot cancel anything there. Which is why free-body diagrams are the central technique rather than a study aid: drawing the boundary makes the third-law partner automatically absent.