I understand the phenomenon I want to explain is something known as Earth Shine. What is Earthshine and what causes Earthshine? Is it really normal? What exactly is the less illuminated part in the diagram?
Yes, earthshine is completely normal. It is simply illumination of that part of the moon that is facing away from the sun. This illumination is sunlight that is reflected towards the moon by the Earth. Strictly speaking, it is that reflected light from the Earth that is itself reflected back to Earth so that we can see it. The intensity of Earthshine is highly variable. If the moon was almost in line between Earth and sun then an observer on the dark side of the moon would see the the whole of Earth's disk brighly illuminated by the sun, and hence the earthshine would be about at its maximum. Conversely, when the Earth is nearly between Moon and Sun, an observer on the moon would only see a crescent Earth (apart from man-made city lights), so the Earthshine would be at a minimum.
You can think of it as the "reverse" of moonshine. When you walk outside at night and there's a full moon, you can actually see quite a lot around you.
Same thing - this happens on the moon when there's a "full earth" that lights up everything around. And since you're on earth, you see this from the outside, on the moon.