The full moon, by legend, corrupts the minds of men, driving even the most logical into fits of insanity. A man of science may look at the moon and consider the brave men who set foot on the hallowed ground for us to expand the reach of mankind toward the stars. In ancient times, they attributed the power of the moon to the gods, even perhaps being a god themselves. Perhaps when a dog looks toward the moon they see Laika, the first dog in space. Whatever it is they see amongst the stars can often make them lose their minds!
The Root of the Behavior
Canines are descendants of their larger counterparts, the wolves. The same behavior, perhaps famously, can be observed in their predecessor and this could indicate that is hard coded in their biology. This correlation can often be misleading. Wolves do not howl at the moon, however they do howl to communicate. As they are nocturnal, they are usually awake and alert at night, and therefore at night is when they do most of their communicating. Canines have a variety of senses that are heightened at night, even if they are not a nocturnal species themselves. For instance, dogs can hear better at night because there is less ambient noise to drown out the sounds around them.
This does not speak to the full moon specifically but just note that your dog may howl at night simply because he hears the howls of the dogs and wolves around him, even if you cannot, and is just wailing out their reply.
If it happens specifically during the full moon, it may speak more to what they can see. At night, animals come out to play. All of the animals who need to hide in the cover of night from their predators come out during the night and indeed your dog would probably be one of those predators without human intervention. During the full moon, they are basked in the light of the night sky and all these critters become visible for your dog to see and then immediately lose their mind. There is a decent chance that your dog just may be a bit weaker in spirit than you expect! The light and sounds that emanate from a well moonlit night that may just push your dog into fear, and the barks and howls are simply the natural reaction your dog has to any fearful situation.
Encouraging the Behavior
If your dog likes to go out at night and yelp at the moon to the dismay of all your family and neighbors, what can you do? There are certainly a few things you can do about this if it is a burden, but note it has no health or safety risks for your dog, so if it does not bother anyone, it might not be too important. Assuming it is a burden, the logical solution would be to make sure he is inside on those nights. Remember your dog is not necessarily barking at the moon, instead at the animals illuminated that otherwise would not be visible. In this case, there may not be a ton you can do. It's like a squirrel running across your fence. If your dog likes squirrels, he is going after it. Through specific training regiments you can remove the end behavior, but not the urge. In the case of the full moon, it is hard to get the redundancy needed to remove the reaction through training. You can also pest control your yard. Make sure you don't have rabbits running about and other vermin that your dog could be yelping at. Try and make sure your dog has a safe and comfortable environment they can retreat to so you can eliminate the fear provocation. Play some music to drown out the howls of other dogs and wolves and you may see your dog respond much less.
Other Solutions and Considerations
The additional sights and sounds of the evening can make your pup go nuts. Reducing this stimulus by placing them inside will likely remove this behavior entirely. If they are staying out during the full moon, then they will be in a world of wonder as the moonlit sky reveals all sorts of critters they would not normally ever see. All the dogs in the neighborhood will get that experience and even if your dog would not normally howl at the moon, your neighbors dog might. Like a yawn in the subway it'll get passed around. The analogy being if your neighbors dog does, then chances are your dog will as well in some sort of reply.
Maybe the mystical powers of the full moon really do exist, and maybe it contributes to the insanity your dog displays that night. Regardless of that sorcery, other factors definitely come into play, and those factors can be controlled and influenced to reduce the howling at night. A trainer or behavioral specialist will be able to give you more insight about your dog and recommend your best steps moving forward.