Surly the moon is not projecting a direct beam of light which only lights up the small area of clouds around the moon. What phenomenon is happening that only allows us to see that center focus glow and not all the clouds glowing?
This is an experiment that you can do at home:
- Put a paper fixed in front of a light bulb
- Move your head around and see how the bright spot on the paper accompany your moves.
But, why is that? How is it that not all the paper is lighted the same?
When light travels through a translucent material some of it gets reflected back, some bounces inside of it and some goes through. As always the shortest path is the easiest, so the light coming straight "bulb-paper-eye" gets through easily (brightest spot) and as you go further from the center you are not seeing direct light anymore, but diffused light, so it gets dimmer and dimmer because to get to your eye it has to bounce in the right direction. That doesn't imply that the rest of the paper is not getting light, it is just not bouncing in your direction and the same happens with the clouds.
The light is scattered by the particles in the clouds. These particles are relatively large, compared to the wavelength of light, and so the type of scattering is called "Mie Scattering".
Light that hits a cloud particle may be scattered in any direction, but it is much more likely to be scattered forwards than than to be scattered to the side. What this means is that around the moon the clouds appear lit up, to the side the clouds appear dark. The amount of scattering doesn't depend much on the colour of the light, so the scattered light appears the same colour as the moon (ie white)
Here is an image from hyperphysics http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html which illustrates how Mie scattering creates a white glare around the sun, or moon, and how Rayleigh scattering from air molecules causes the sky to appear blue.