The sound you hear is a wave that is a sum of all sounds around you. Waves have a few relevant properties: They travel at a known velocity, and they are additive.

To cancel such wave in your ear: we measure the wave just outside the ear and play its inverse with a small delay from the earphone. Notably, this only cancels the sound in a very small region around the inner side of the earphone. Everywhere else it adds its miniscule amount of more sound to the wave.

For best results: You need a good microphone in both earphones, and a good algorithm to slightly alter the wave, to mimic hiw it will be altered by the earlobe (as the in-ear earphone sound is not altered by the earlobe identically to the sound coming from the outside). Fortunately, we can tune this individually: place a second microphone inside each ear canal (near the very tip of the earphone), and measure which delay and which amplitude modifications reduce the sound the most.

A good analogue: Look at the waves in the see. Measure the height of the wave. If it is above the mean water level, push the water down with a paddle you have placed under the surface. If it is below, push the water up. If you move your paddle at just the right speed for a given measurement, you can destroy the wave around your paddle (whilst creating a new wave around your paddle, propagating outwards and adding a bit to the waves everywhere else in the sea).