The festival is celebrated with a big feast where friends and families gather to enjoy mooncakes and kids play with colourful lanterns. The moon, a symbol of wholeness, unity and harmony in Chinese traditions, is the brightest and fullest on that day.
According to the story, there was an uprising in China against the Mongolian rulers during the 14th century. As mass gatherings were banned, planning a rebellion was impossible. Secret messages were then hidden inside mooncakes giving the exact date and time of the revolt against the Mongolians. On the night of the Moon Festival, the rebels successfully attacked and overthrew the government. What followed was the establishment of the Ming Dynasty. To celebrate their victory, eating mooncakes has become an important part of the festival.
You can read more about the festival here and here.
Then there is the Legend of the Moon Goddess and her rabbit here.