Why doesn't Newton's third law make the net force zero?

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llamajuice
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Newton's third law says that for every action force there's a simultaneous reaction force equal in magnitude to the action force, but in the opposite direction. So doesn't that make the net force zero? If that's true how does anything move?
 

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The third law says that the force exerted by object A on object B is equal but opposite to the force exerted by object B on object A. Note very well: Those equal but opposite forces act on different bodies. The acceleration undergone by object A depends only on the forces that act on object A. That object A is also exerting forces on other objects does not affect the acceleration of object A, at least not directly. The connection is indirect.

Suppose A and B are the only objects in an otherwise empty and very large region of space. Whatever other forces are acting on A and B from outside that large region of space is so small that it can be ignored. A will accelerate because B is exerting a force on A, and B will accelerate because A is exerting a force on B. The nature (magnitude and direction) of these forces will change over time, so what A does to B now will change what B does to A in the future.