When you see 'heat rising', what are you actually seeing?

Warmer air has a different density to colder air, so the light refracts/bends, like through a prism.
The shadows occur as the light happens to have been focused away from those points.

Edit: Not a scientist. Currently high.

https://en.wikipedia.org/wiki/Refraction#Atmospheric

Hot air is less dense than cold air, and the speed of light through a fluid will change with the fluid's density. Heat shimmers are a swirling mass of air at different temperatures.

https://en.wikipedia.org/wiki/Convection#Natural_convection

Light that would have hit the shadowed region instead got sent in a different direction.

In a not super scientific explanation, hot air rises and is less dense than cold air, this changes how light passes through it compared to the rest of the air. Shadows are essentially less light dense areas, so by changing how light passes through it you naturally get less dense regions behind the distortion making it darker than the other areas, aka a shadow. Air, being transparent though, there is less distortion than an opaque object, such as yourself, which has greater ability to distort (or rather, prevent) light in that area. (Your shadow)

You’ve gotten some good answers explaining that heat changes the density, and therefore the index of refraction of air.

Fun fact: Schlieren Imaging allows one to photograph shockwaves by relying on the same effect. As a shockwave travels through air, it creates a region of high density, which can be imaged with this technique.

this is a 6 minute video on the process for photographing changes in air pressure, it should answer a lot of your questions