The Japanese formula means, roughly, homage to Amida Buddha, and the entire theological weight of Pure Land Buddhism sits on what saying it does.
The practice descends from Chinese nianfo, attested from the fifth century, which in turn rests on the Larger Sukhavativyuha Sutra and its account of the bodhisattva Dharmakara, who vowed not to accept buddhahood unless beings who called on him with sincere mind were born in his land. The eighteenth of those vows is the one the tradition builds on.
The Japanese development is the interesting part. Honen, in the twelfth century, took the position that in an age of decline the disciplined practices, meditation, precepts, scholarship, were beyond most people, and that exclusive recitation of the name was the practice the Buddha had provided for exactly that situation. He was exiled for it. Shinran, his student, pushed further: if birth in the Pure Land depends on the vow rather than on the practitioner, then reciting the name as a means of earning it reintroduces the self-power the whole position rejects. On Shinran's reading the recitation is gratitude for something already assured, not a technique that secures it, and a single utterance in faith suffices.
This is structurally very close to a debate Western readers know from a different religion entirely, about faith and works, and the comparison has been made often. It is a comparison, not a genealogy.
Shinran also married and had children, and the tradition he founded became the largest Buddhist denomination in Japan and one with a married clergy.