An easy way to see this one is with a lamp and a ball. The lamp represents the Sun, and the ball is the Moon. Your head is the Earth. We want to simulate night, so the lamp should be behind you.

Hold the ball in your right hand and hold your right arm straight out in front of you. You want to hold it high enough that the light from the lamp hits it. This is like a full moon. The light hits the ball (moon), and reflects back towards the Sun and towards you (Earth).

Next, move your arm straight out to the side. Looking at the ball, you will see that the side facing the lamp is brightly lit, and the other side is darker. This is like a half moon. Even the lit part of the ball does not look as bright, because much of the light is still reflecting back towards the lamp (Sun.) Since you are no longer in between the ball (moon) and the lamp (Sun), you get much less light.