I'm guessing what you see is the moonlight being scattered by the hexagonal ice crystals in cirrus or cirrostratus clouds, which lie at very high altitudes, 5-6 km and above. The light is scattered by roughly 22º, and because of the slight wavelength dependence, the halo actually has rainbow-like colors, although often they are so faint that you just perceive it as white.
Read more here.
EDIT: My first was drawing was made with my head under my arm, and had the angle wrong. Rob Jeffries made me aware of this, and also pointed out that larger angles are possible, up to 50º. Moreover, I drew the colors in the opposite order. Since blue light is deflected slightly more than red light, the blue light that reaches you is deflected from the part of the clouds that are more distant from the Moon.
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