Planting by the moon is one of the oldest forms of gardening, and is the least backed up by science. Other than the seasonal rules, it has the biggest numbers of followers that swear by the methods. The only overriding rule they follow is the seasons. These though can be also be controlled by star signs the moon falls into, although these are not universal as the north and south hemispheres differ.
What we do know is the moon affects the tides, people's moods, and certain plants will flower according to the moon cycle. Hence the belief that the water and sap move up the plant better due to gravitational pull from the moon. Statistically there is 2 odd percent more likely and there is more volume of rain during the increasing moon.
What works against it, is that the moons gravitational pull does not affect rivers or lakes, as these bodies of water are too small meaning that it has little to no effect on plants.
The old Indian farmers used the moon to plant, as the heat of the day was too much, and here in lies my reasoning shy the planting plan does work. With the new moon and added light, the seeds for above ground were sown and transplanted as they could be watered and looked after every night while the root crops were planted in the waning light as they go under ground and are not that water critical. Transplanting at night let the plants settle for the night before the heat of the day.
Planting by the moon works as the methods are organized and structured. You plant and water, look after, weed and feed at regular intervals. The moon is the reminder. So while the moon may not be the benefit, the method of planting by it has merit. Other things like planting in the overcast rainy days are more beneficial, but they cannot be planned. I think my plans work so will not change, but for those that need some formal structure, this is certainly a new age way of ancient practices that will guide you.
Let me know your thoughts, these are mine and based on reading and watching many different views on the subject. There is not much science to prove the methods right or wrong.