It's a little bit less magical than what people are suggesting here.. You don't actually hear as well, so to speak, as a microphone. So there's enough time to invert the soundwave and play it back before you start picking up the vibrations that produce sound you hear.

Alternative way to think about it - you delay the incoming sound slightly and then play it back as perfectly out of sync as you can. The question really is the response, and how quickly you can generate the wave accurately.

The trick is that you should be producing a sound-wave that sounds like what is actually heard behind the clogs, for example. And you really don't want to play back a really, really loud sound, or increase the wave too quickly based on some extrapolation, etc. And it's typically not perfect, so you get noise. You can also mask it all and increase response, so to speak by having a noise-floor.

But yeah, if you play back some fairly low volume sound where the noise is not physically noticeable, and you allow for some noise on the bottom here -- an exactly out of sync wave is going to cancel the sound out, in the sense that your ear is not going to vibrate and make you hear sound.