A lot of fish are nocturnal feeders, like sharks for example. The best visibility for a night feeder, would be during the full moon. It is simply easier to catch a fish at night, when the bait is visible. This applies to a lot of bottom feeders as well. I have heard that the snapper bite is best on a full moon many times. For many species of fish, including snapper, spawning occurs during a full moon. This could explain why some of those best fishing days occur on the full moon.
For the new moon, since there is less light at night, the fish are simply more hungry during the day. Some pelagic fish use their eyes more than nose to find food. During the new moon, there is very little light and they have a harder time finding bait/food in the darkeness. It would make sense that they do not eat as much on new moon and feed heavy during daylight hours.
The moon phases also affect the tides. You will hear many fishermen who love to fish the incoming or outgoing high tide. I think it is hard to say that the moon phases do not influence your fishing. What I do know is that every May, 3 days before and after the new moon, it has been excellent fishing off Hatteras. I plan my trip based on the moon phases that month for sure.