Why does the moon change shape?

No — the earth’s shadow has nothing to do with it. The half of the moon facing the sun is always lit. What changes is how much of that lit half we can see from earth as the moon moves around its orbit, and those are its phases.

At new moon the moon lies between earth and sun, so its lit side faces away and we see almost nothing. At full moon the earth is in the middle and the lit side faces us completely. In between we see part of it — a half or a crescent.

The earth’s shadow does fall on the moon during a lunar eclipse, but that does not happen every month — the moon’s orbit is tilted slightly against the earth’s, so at most full moons it passes above or below the shadow.

The moon always shows us the same face, because it takes as long to turn on its axis as to orbit the earth. That makes “the dark side of the moon” a misleading phrase — that side is hidden from us, but it gets just as much day and night as the side we see.