The idea is somewhat simple, you simply measure (and in more expensive/sophisticated headphones predict) the noise coming into the ear and use that info to send noise into the ear which exactly counteracts the incoming noise you want to block.

There are several insanely difficult parts to this though:

1. How do we calculate what we need to counteract incoming noise fast enough that we actually counteract it in time? In airpods there may be a couple centimeters between the outer speakers measuring incoming noise and the inner speakers. This gives distance/speed of sound ≈ 3*10^(-2)/343 ≈ 1/10000 seconds, so 100 microseconds. That's not a lot of time to calculate things.

2. The noise being measured at the outer speakers is distorted due to the shape of your ear. This needs to be taken into account or otherwise the inner speakers would not be cancelling out the correct sound.

3. Predicting what noise will be coming and using the prediction instead of actually measuring the noise. This works pretty well for a lot of signal processing purposes (separating noise from music, recreating ECG's when only part of the data is available, or anything else where the input can be considered a signal) but real life noise is not always predictable.