Active noise cancelling is placing between ears and source an inverted wave generator.

A soundwave is coming, you intake it with a microphone and attempt producing its inversion in real time in speakers, to cancel it.

As I have written, it may not work properly with impactive noise - it can do the opposite, because the impactive sound is cleaner. Or, it may be calibrated for impactive noise - so there exist earmuffs that recognize shooting sounds at their inception, to reduce these selectively.

They're like shooting muffs/cans, except they have a microphone on each side, batteries, and an amplifier. when it's off, they're regular hearing protection. When you turn them on, you can adjust the volume from "slightly lower than without the muffs on", to "actual volume", to "hear a cricket fart across the street". When a loud sound is detected - say, clapping, gunfire, hitting an anvil - it shuts off the amplifier.

As an aside, since your ears are not ringing / recovering from a very loud sound, and the amplifier turns back on very quickly, you can hear the reverberations, echos, and sympathetic resonances. It's quite cool.