The correct explanation for why the moon appears half lit and half dark from the surface of Earth on two days each month is:
Half of the moon's surface that is illuminated by the sun is visible from Earth's perspective.
This occurs during the first quarter and last quarter phases of the moon, when we see half of the sunlit side of the moon facing us. The other options describe different phenomena, such as a partial lunar eclipse, which is not the cause of the moon appearing half lit. In this case, the moon is simply in a position where we see one-half of its illuminated surface.