What we see is light being refracted in "weird" ways. It's similar to the way light above a very hot surface, like asphalt on a hot sunny day, appears to be warped.

Everything has a refractive index, meaning a number that determines how much light is warped. Air is usually very evenly distributed, so you don't see much (or any) warping. But the airplane moves so fast that it pushes the air in front of it to the side, which means around the airplane the air is denser. The refractive index of air changes with density. This makes the light appear so warped :)

Fun fact, the sonic boom also comes from the air being pushed to the side. Behind the airplane there is a pocket without air (or with very little). The higher density around the airplane is not stable, gases want to be evenly distributed. This makes the air crash violently into itself as it fills the pocket, which produces the sound! This also explains why you hear the sonic boom everywhere along the path of the airplane, and not just where it breaks the sound barrier.