Lions mostly hunt in the dark, as predators have more of an advantage if it is as dark as possible. Hence, the moonlight is a disturbing factor, because it makes the predators easily detectable by their prey. And when the full moon is actually in the sky, it then so happens, that the lions occasionally come away empty-handed. Thus commences a time at full moon (and shortly before), where the lions begin to starve and hence their urge to hunt increases.
After the full moon, the moonlight diminishes and the Moon rises every night a little bit later, which creates a time span after sunset where there is no moon light in the sky and this benefits the hunt of the lions.
Scientist have researched lion attacks on humans in Tanzania and assessed that the likelihood of an attack is many times higher within the first 10 days after the full moon, than in the 10 days before the full moon (lions are still full from their more successful hunts at the new moon).
Packer C, Swanson A, Ikanda D, Kushnir H (2011)
Fear of Darkness, the Full Moon and the Nocturnal Ecology of African Lions.
PLoS ONE 6(7): e22285. doi:10.1371/journal.pone.0022285
For the sake of completeness, let’s clarify again that the lions are not being masterminded by “magic moon powers”. Their hunting behavior is simply influenced by the amount of light. Pure physics, actually …