Effortless is right. The hifi demonstration is good, although, you'll find that the effect differs depending on where you are in the room. The lower the bass, the longer the wavelength, therefore larger "patches" of destructive interference.
Another example is ripples on a pond. Drop two stones into a pond and when the waves collide they add to each other. Two peaks colliding makes a really big wave (at that point) two troughs make a really deep trough (at that point). ANR tries to guess where your ear is and what the sound wave will be at that point (peak or trough) then it adds the opposite sound.
Oh dear! I'm making it even more complicated!!
Short answer: Yes, ANR does remove the sound around your ears. It's not masking it.