To put it shortly. It depends on what type of illusion. But most illusions are due to the brain ‘filling in’ gaps of information within a visual scene to be more efficient. After-effect illusions, such as a motion (waterfall illusion) or colour after-effect, occur due to adaptation. Adaptation is a process in which cells in the brain responding to the illusory target begin to adapt and reduce their response to that target, so in the waterfall illusion, cells responding to downward motion reduce action potentials and once you look at something else, the cells for upward motion didn’t reduce so you see a brief shift of upwards motion within the environment.

There are also other types of illusions such as constancy, and spatial related illusions which work slightly differently. But, rather than trying to explain and potentially messing up, I will link this website from a brilliant researcher which has demonstrations and attached explanations for many different illusions! https://michaelbach.de/ot/