From Slate recently:
What does the fullness moon have to do with it? Doesn’t the moon do it’s thing all the time?
It’s the combination of the sun and the moon and their relative positions during the various phases of the moon. When the moon is full, it’s opposite the sun. When it’s new, it’s on the same side of the earth as the sun is. Has nothing do with the light from a full moon. 
Think tides. “Spring tides” (unrelated to the season) occur during the new and full moon phases during which the gravity from both the sun and the moon are pulling along the same axis. The high tide is higher and the low tide is lower on these days because of this. The tunnel of the particle accelerator is so large that it is experiencing tides, and the distortion is the largest near the full moon due to the larger difference between the low and high tides that the tunnel is experiencing.