In the destructive interference picture you can see as one wave goes up, the other opposing wave goes down, creating a sum of zero. Think of a vessel with water and a pump. If the pump is pulling water out of the vessel it drains quickly. If another pump pushes water in, the vessel fills. If both pumps push and pull at the same time and in equal measure, the water level never moves. The push of one pump cancels the pull of the second pump.
That is how active noise canceling works. One of the challenges of mechanizing such a scheme with loudspeakers is the microphone will also hear that which the speaker produces as well as what's around it, a problem not encountered when using headphones.
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