The idea of 'intercession on the part of the priesthood for the benefit of souls in hell' is antagonistic to the explicit teaching of early Buddhism. The origin of the custom is unknown, but it is attributed to Śākyamuni, whose disciple Maudgalyāyana is represented as having been to purgatory to relieve his motherʼs sufferings. Śākyamuni told him that only the united efforts of the whole priesthood 十方衆會 could alleviate the pains of the suffering. The mere suggestion of an All Souls Day with a great national day for the monks is sufficient to account for the spread of the festival. Eitel says: "Engrafted upon the narrative ancestral worship, this ceremonial for feeding the ghost of deceased ancestors of seven generations obtained immense popularity and is now practiced by everybody in China, even by Daoists and Confucians." All kinds of food offerings are made and paper garments, etc. burnt. The occasion, 7th moon, 15th day, is known as the 盂蘭會 (or 盂蘭盆會 or 盂蘭齋 or 盂蘭盆齋).

In Japan, this is a popular ritual known as the Bon festival—obon 御盆, お盆, or the 'ghost festival.' In ordinary Japanese usage, urabon is shortened to bon, and the honorific prefix 'o' {o 御, お) is added. Although the Ullambana Sūtra itself became popular in China starting from the sixth century and helped the Buddhist saṃgha there establish itself as a participant in indigenous modes of ancestor worship, there was some Indian precedent for the idea of dedicating merit earned by supporting the saṃgha to help ancestral spirits. Scholars debate the etymology of the Buddhist Sanskrit term ullambana, but its derivation remains obscure. One theory is that it comes from the Sanskrit avalambana, meaning 'hanging upside down,' a possible reference to the pitiful state of spirits who are 'left hanging,' as it were, when they have no living descendants to make the usual ancestral offerings of food and drink to them. Another theory traces the etymology of ullambana to uruban, a Persian word for spirits of the dead. A folk etymology is that ullambana refers to 'bowls' (C. pen 盆) that are used in making offerings to spirits, but the Chinese character pen 盆 was probably used simply for its sound value in transliterating the third syllable of ullambana. Seishi Karashima, who has investigated the origins of the term in his 2013 article, reports:

It is, therefore, probable that yulan 盂蘭 (MC. ju lān) is a transliteration of a Middle Indic form *olana (not attested) from Skt. odana, and yulanpen 盂蘭盆 may mean 'a bowl for boiled rice,' i.e. 'a rice bowl' . Odana in this compound was probably transliterated in order to designate that this was no ordinary rice bowl but one specially used for the ceremony. Consequently, the original title of this scripture, Yulan jing, might mean “Odana Sutra” or “The Sutra of Boiled Rice”, while Yulanpen jing could be “The Sutra of Rice Bowl”. (pp. 301–302)

The Ullambana Sūtra makes the case that the traditional Chinese mode of ancestor worship, which involves 'giving nourishment'  供養 to the spirits by placing offerings of food and drink on an altar, may not succeed if the bad karma of the ancestors themselves has resulted in their rebirth as hungry ghosts. The sutra illustrates this point with the story of the monk Maudgalyāyana 目連 mother, who having been reborn as a hungry ghost, is unable to consume the food offerings he gives her: whatever she lifts to her mouth to eat bursts into flames. To be truly filial, the sutra argues, one should first make donations to the saṃgha of monks and nuns, the most fertile field of merit, thereby tapping into a huge store of good karma that can be used to force the offerings through to the ancestors and help them into a more happy state of existence. As the Buddha Śākyamuni tells Maudgalyāyana in the sutra:

…on the fifteenth day of the seventh month, the day on which Buddhas rejoice, the day on which monks release themselves, they must all place food and drink of one of the hundred flavors inside the yulan bowl and donate it to monks of the ten directions who are releasing themselves. When the prayers are finished, oneʼs present parents will attain long life, passing one hundred years without sickness and without any of the torments of suffering, while seven generations of ancestors will leave the sufferings of hungry ghost-hood, attaining rebirth among gods and humans and blessings without limit. (Translation by Stephen F. Teiser, The Ghost Festival in Medieval Japan [Princeton, N. J: Princeton University Press, 1988], p. 53).

The Ullambana Sūtra informed the practice of feeding hungry ghosts 施餓鬼, also known as 'saving the burning mouths,' which grew tremendously popular from the eighth century on and enabled the Buddhist saṃgha to associate itself with traditional Chinese modes of ancestor worship. This helped deflect a major criticism of Buddhist monks in medieval China, which was that they were unfilial because, as celibates, they produced no descendants to care for their ancestors. Through the ghost festival, the saṃgha was also able to promote itself as a kind of public charity organization that could care for and placate disconnected, potentially dangerous spirits who had no family, thereby protecting the imperial state and the populace at large from their baneful influence. The feeding of hungry ghosts also expressed the Mahayana Buddhist ideal of universal compassion and sent the message that the 'family' of the Buddha included all living beings. The traditional date for the Bon festival is the 15th day of the 7th month by the Chinese lunar calendar, and Standard Observances of the Sōtō School honors that tradition by giving July 15 as the date of the Bon festival great food-offering assembly (urabon daisejiki e 盂蘭盆大施食會). Because Japan adopted the Western (Gregorian) calendar in modern times, however, in many parts of the country Bon is celebrated during the week centered on August 15, which feels closer in season to 7/15 by the old lunar calendar. In popular Japanese belief, Bon is the time when ancestral spirits 'return' to visit the world of the living and should be greeted with due respect. People clean the graves of family members at this time and make offerings of fruit, fresh flowers, and incense. They invite Buddhist priests to their homes to perform sūtra chanting services in front of family buddha altars 佛壇. Because the spirits need guidance through the dark, lanterns or candles are sometimes lit on graves. In some communities, candles are sometimes placed in paper boats and set adrift in rivers, and in the city of Kyoto a vast bonfire in the shape of the Chinese character 'great' (dai 大) is lit on a mountainside. During the week of the Bon festival many Japanese Buddhist temples, including those affiliated with the Zen schools, hold assemblies for feeding hungry ghosts 施餓鬼 at which an altar for the 'myriad spirits of the three realms'  三界萬靈 is set up and the ritual cycle known as the Ambrosia Gate (Kanromon 甘露門) is performed by a group of monks. The merit produced in that rite is dedicated to the spirits of deceased family members of the parishioners who attend. Each family is typically given a new stūpa board 塔婆 to place next to the family gravestone.

Reference:

Karashima, Seishi. 2013. “The Meaning of Yulanpen 盂蘭盆 — 'Rice Bowl' on Pravāraṇa Day.” Annual Report of The International Research Institute for Advanced Buddhology 16 : 288–305.

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