A book I've been skimming through recently, How to Grow More Vegetables and Fruits, talks about this, and I've also heard it mentioned, and read about it several other places--perhaps most notably the Farmer's Almanac which I have seen for sale on many news stands, and which has a web site.
From the book:
One of the most controversial aspects of [our] method is [the] method of planting seeds and transplanting seedlings according to the phases of the moon.
... Short-germinating and extra-long-germinating seeds are planted 2 days before the new moon, when significant magnetic forces occur, and up to 7 days after the new moon. Long-germinating seeds are planted at the full moon and up to 7 days afterward. Seedlings are transplanted at the opposite time.
(emphasis in original)
It goes on to explain in much greater detail when certain types of seeds and seedlings ought to be planted or transplanted, in accordance with lunar phases.
The gist of it seems to boil down to the idea that moonlight helps leaf development, and that lunar gravity, helps root development. (I'm not really sure what it means by the "magnetic forces" in the above quote).
At least the book acknowledges, right off the bat, that this lunar stuff is controversial. But my question is: Is there real evidence that there is a benefit to plants to be planted and/or transplanted by phases of the moon?