This beautiful phenomenon is called Earthshine. The soft glow on the shadowed part of the moon is caused by the reflection of sunlight from the Earth.
Specifically, Earthshine happens when the light from the Sun is reflected from the Earth’s surface, to the moon, and then back to our eyes. Because of this double reflection of light, Earthshine is many times dimmer than the direct light of the Sun on the Moon.
Earthshine can be best seen during the crescent phases (the 1-5 day period before or after a New Moon).