Why is it dark at night?

For the same reason why it's dark in shadows - there is little to no light.

The half of the Earth facing the Sun, is what shadows the half of the Earth facing away from the Sun.

It's dark at night because night-time means you're in the Earth's shadow.

Because the sun sets; the earth is spinning but the Sun isn't. As the Earth spins different areas of land receive light and the rest dark. This is also connected with the changes of time in different countries.