Why should a shadow be cast on something?
Look at this picture taken at sunset:

What is that dark blue band at the horizon?
Answer: the shadow of the Earth itself:

In a certain way, we can say that the night sky itself is the result of the shadow of the Earth being cast on the sky (think about it).
Basically, when you are standing on top of the mountain, you are seeing a portion of the sky in which it is already night. The effect is even clearer in other pictures, like this one:
The sky is not completely dark in the cone of shadow because of scattering from the atmosphere surrounding the cone, so it retains that orange-ish or blue-ish color, depending on the hour the photo was taken.
I include a (totally professional) sketch illustrating the phenomenon:

Update: the role of the atmosphere
It looks like that my answer was a bit too synthetic and the role of the atmosphere was not so clear. This is my fault, so I would like to clarify this point.
The presence of the atmosphere is fundamental for the phenomenon. In fact, if the atmosphere was not present, we would see a pitch black sky, as we all know having seen photos taken from the moon. The atmosphere, and in particular the phenomenon of Rayleigh scattering, is responsible for the color of the sky.
What happens basically is that air molecules hit by light rays from the sun scatter it with an intensity proportional to the inverse fourth power of the wavelength, $I \propto \lambda^{-4}$: since blue-ish has a shorter wavelength than green-ish, red-ish etc., the sky results blue (so why are sunsets red?).
The portion of the atmosphere in the shadow cone of the mountain is not hit directly by the solar rays, but only by light scattered by the atmosphere outside the shadow cone. This means that the color of the atmosphere in the cone will be quite the same as the color outside the cone, only much darker.
So, in a certain sense, we can say that the shadow is being cast on the atmosphere itself.
My rethorical question (Why should a shadow be cast on something?) was perhaps misleading, but I only meant that the shadow was not cast on a solid object or on a cloud.