Sounds can be drawn as a wave, so for one of the simplest examples let's look at a sine wave - just a simple, regular wave shape.

As a simple maths experiment, if we mirror this wave around the horizontal axis so that where one wave goes up, the other wave goes down, and then add them together, they will cancel each other out.

This works with sound in headphones too. If we stick a little microphone on the outside of your headphones so that it detects the outside sound you will hear, then takes that sound and inverts it so it is upside down and plays that through your headphones, the two will cancel out and the outside sound will vanish. If you play the inverted sound alongside the music or other audio you actually want to hear, you get both at the same time - no outside sound, just the extra audio you are adding.

Incidentally, this works really well with headphones, because the microphone can easily detect exactly what your ear will. If you tried this with a whole room, the way sound reflects off of surfaces will mean that the 'noise' sound will be different depending on exactly where you stand, so it is impossible to cancel it out - you could set it up to cancel nicely in one place, but if you moved about in the room in other places it won't work at all, or may make the noise worse.