I have talked with many captains and fishermen over the years, and some swear by moon phases, while others say they have little effect. What I do know is that there are more world record fish are caught during new and full  moons. But, this kinda makes sense, regardless if you want to argue about correlation/causation of the statistics.

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. For the new moon, since there is less light at night, the fish are simply more hungry during the day, because they have a harder time finding food in darkness. 

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 in. I plan my trip based on the moon phases that month for sure.