Why the Moon shows phases

The Moon shows phases because it shines by reflected sunlight and we see a changing fraction of its lit half as it orbits. Half the Moon is always in daylight, and the phase is only how much of that daylight side faces Earth.

The dark part of a crescent is not the Earth's shadow, which is a lunar eclipse and a rarer event. Ordinary phases are the Moon's own night, the hemisphere turned away from the Sun, seen at whatever angle this month's geometry allows.

The mechanism is more particular than the slogan. At new moon the lit half faces mostly away from us, so the Moon sits near the Sun in the sky and we look at the night side. About two weeks later the Moon is opposite the Sun, the lit half faces us, and we call it full.

Follow it one step further. In between, we see a slice, a half, then a gibbous bulge, as our viewpoint swings around the ball of rock. The cycle from new moon back to new moon takes a bit under thirty days, longer than one pure orbit relative to the stars, because Earth has moved around the Sun in the meantime.

The case worth keeping in mind is specific. A thin crescent hangs near the sunrise or sunset glow, and you can often make out the rest of the disc faintly lit by sunlight reflected from Earth. That earthshine is a second, dimmer illumination, not a failure of the phase to be mostly night.

That case has a boundary. Phases do not require the Moon to change shape, glow by itself, or pass behind a fixed dark filter. Clouds, and the simple fact that a new moon is up in the daytime glare, hide many of the positions, which is an accident of viewing rather than a gap in the orbit.