Why do fish-eye lenses produce distortion near the edges of an image?

Fish-eye lenses produce distortion near the edges of an image because of the way they manage light rays to achieve an extremely wide field of view, typically ranging between 140° and 210°.1

For image angles greater than 110°, it becomes physically difficult to position the lens close enough to the film or sensor to allow light rays to diverge sufficiently. To overcome this, the fish-eye lens is designed to make the rays diverge less behind the lens than they do in front of it. This optical configuration results in appreciable distortion, where image details near the edges and corners of the frame appear progressively compressed. While this effect is often considered a distortion, it is frequently utilized as a deliberate pictorial element for wide-angle effects or in specific scientific applications, such as recording horizon-to-horizon cloud formations.1