The Ghost Festival is called the Festival of Ullambana (Yulanpen in Chinese, literally meaning rice bowl), based on the Scripture of the Ullambana Bowl (Yulanpen jing), translated by Dharmarakṣa in the third century. This festival is significant for Buddhism to take root in Chinese society by focusing on the traditional Chinese value of filial piety.

In traditional China, Confucian political ideology advocated a parallel structure in political and social structure surrounding filial piety. The ruler as the Son of Heaven represents Heaven as the father of all the ruled people. The ruler should be benevolent, and his people should obey him, serve him, and serve the empire. The local government officials served as parents for taking care of their people as their children. The people paid taxes and offered services. A father took care of his family in exchange for his family’s respect and obeisance; a son would carry on family tradition. With the spread of Buddhism to China, a new ideology of filial piety developed that focused on the relationship between mother and son: the mother had the innate sin from her reproduction and delivery and would be sent to hell and should be saved. The son could make merit by saving his mother through becoming a monk and supporting the monastic order, fulfilling his filial piety obligation and serving Buddhism.

In early Buddhism, in the Śāriputtattherassa mātupetivatthu (the Elder Śāriputra’s Mother Peti Story), Śāriputra’s mother passed away and became a ghost, afflicted with hunger and thirst, and eating the pus and blood of animals and men. She asked her son to save her. He answered the call by offering four huts with food and drinks to the Sangha of the four quarters, and then dedicated the donation to his mother. His mother received the donations of food, drinks, and clean clothes from his dedication.

In Mahāyāna Buddhism, Mahā-maudgalyāyana (Mulian in Chinese) becomes the leading role in the Buddhist narrative of filial piety. According to the Scripture of Ullambana Bowl, Maudgalyāyana had just obtained the six penetrations and wanted to save his parents to repay their kindness in raising him. He surveyed the world with his divine eyes and found his deceased mother being born among hungry ghosts. She had neither food nor drink.

When the assembly of the monks of the ten directions attend the confession ceremony at the end of the rainy season retreat, one should prepare rice, food of a hundred flavors, five kinds of fruit, vessels for drawing and pouring water, incense, oil lamps, candles, mattresses, and bedding; place the tastiest food in the world in bowls; and offer these to the monks. Then one’s ancestors will be able to escape from the sufferings in the three paths, will be liberated immediately, and will be spontaneously clothed and fed. So the idea is to teach Buddhists that, in order to save one's parents or make them happy, one should prepare rice bowls and offer rice and food to the Buddha and the community (Sangha). This ritual will repay the loving-kindness of their parents, who raised and nourished him or her.
Mulian came to save his mother.
Mulian saw his mother being tortured in hell.
The story of Mulian saving his mother in vernacular literature.

How Ullambana (or the hungry ghost festival) came about? (very brief and concise video) 

Ullambana -- Remembering the Divine Love that Saves Humanity (Chinese and Japanese narration with captions in multiple languages) 

Reading:

Stephen F. Teiser, The Ghost Festival in Medieval China. Princeton University Press, 1988.

Alan Cole, Mothers and Sons in Chinese Buddhism. Stanford: Stanford University Press, 1998.