Sacrificing to the moon
Originally, the festival of sacrificing to the moon was set on the Autumnal Equinox day. According to historical records, as early as the Zhou Dynasty (c. 11th century-256BC), the ancient kings by custom sacrificed to the sun on the Spring Equinox, and to the moon on the Autumn Equinox.
But since it is not a fixed day in lunar August, there might be no full moon on the Autumnal Equinox. During the festival, if there was no moon to make sacrifices to, it would spoil the fun. Thus, the day was changed to the Mid-Autumn Day.