The interpretation of a halo around the moon as a sign of impending rain (or snow) stems from:

Seeing those icy clouds refracting or bending light around the moon means cirrus clouds are present, which might indicate a coming storm. If you see a halo around the moon on a clear night, it could be an indication that bad weather is moving in. you're actually just seeing the light of the moon refracting through ice crystals. They do precede some low-pressure systems by a day or two, and low-pressure systems do bring precipitation storms