The vow Manu made (why Vishnu chose Dasharatha's house)
In the Ramcharitmanas Svayambhuva Manu and Shatarupa give up the kingdom, do tapas at Naimisharanya on the Gomati for thousands of years on water and then air, and when Vishnu appears with Lakshmi ask for a son like him; there being none like him, he promises to be their son himself when they are reborn as the king and queen of Avadh. Valmiki's reason is Ravana's boon, which omitted men, so the gods ask Vishnu to be born as a man and he chooses Dasharatha.
Shiva tells Parvati, and Yajnavalkya tells Bharadvaja, the first reason for the Lord's descent: Svayambhuva Manu and Shatarupa, the first couple, whose children were Uttanapada, Priyavrata and Devahuti.
Manu grows old, sees that his life has gone to the world's business, hands the kingdom to his son and goes with his wife to Naimisharanya, where they bathe in the Gomati and sit with the sages.
They repeat the twelve-syllable mantra, eat roots and leaves, then live on water for six thousand years, on air for seven thousand, and stand on one foot without breath for ten thousand; Brahma, Shiva and Vishnu offer boons and they do not stir.
A voice tells them to ask; they ask only to see the Lord's form.
The Lord appears, dark blue, lotus-eyed, with crown, garland, bow and quiver and yellow silk, the Goddess at his left; they fall at his feet and he lifts them.
Manu asks for a son like him; the Lord says there is none like him, so he will himself be their son. Shatarupa asks for devotion, love, knowledge and a pleasing life, and is granted them.
He tells them to enjoy Indra's city for a time and then be born as the king and queen of Avadh, where he will come as a man with his brothers; in time they are born as Dasharatha and Kausalya.
After the soratha at 141 Yajnavalkya names Svayambhuva Manu and Shatarupa, the first couple, and their line (Uttanapada and Dhruva, Priyavrata, Devahuti and Kardama, Kapila). Manu, grown old, makes his son king and goes with his wife to Naimisharanya; they bathe in the Gomati and sit with the sages (142). They chant the twelve-syllable mantra, eat roots and leaves, then only water (143); six thousand years on water, seven thousand on air, ten thousand on one foot without breath; the Trimurti offer boons in vain (144). A voice from heaven bids them ask; they ask to see the Lord (145). He appears, dark, lotus-eyed, moon-faced, with crown, earrings, garland, bow, quiver and yellow silk, the Goddess at his left; they fall at his feet (146 to 147). He lifts them and bids them ask (148). Manu asks for a son like him; the Lord replies that there is no one like him and he will himself be their son (149). Shatarupa asks for devotion, love, knowledge and a pleasing life (150). He sends them to Indra's city, then to be born as rulers of Avadh, where he will come with his brothers (151). Shiva told this to Uma; Yajnavalkya turns to the next reason (152).
While Rishyashringa performs the Putrakameshti the gods gather above the altar and complain to Brahma that Ravana, protected by Brahma's boon, oppresses sages, yakshas, gandharvas and the guardians of the world (1.15.6 to 11). Brahma recalls that Ravana asked for immunity from gods, gandharvas, yakshas and rakshasas and left out men out of contempt, so a man can kill him (1.15.12 to 14). Vishnu arrives on Garuda with conch, discus and mace (1.15.16 to 17); the gods ask him to be born as Dasharatha's son by his three queens, who are modesty, fortune and fame embodied (1.15.19 to 21); he promises to kill Ravana with his kin and to live among men eleven thousand years, divides himself in four and chooses Dasharatha (1.15.28 to 31). Brahma then asks the gods to father helpers: Indra fathers Vali, Surya Sugriva, Vayu Hanuman, and the monkeys gather in the forests (1.17).
What is different here:
No Manu, no vow; the cause is Ravana's incomplete boon and the gods' plea, and the choice of Dasharatha is Vishnu's own at the moment of the sacrifice.
Sources: Tulsidas, Shri Ramcharitmanas, Bal Kand, dohas 141 to 152 and 186 to 198 (Gita Press Hindi edition with commentary, archive.org; Bharatkosh doha pages; the Ram Janam Stuti); Manu Satroopa Tapsthali, Naimisharanya (Sitapur district administration, Uttar Pradesh); Valmiki Ramayana, Bala Kanda sargas 8, 9, 14, 15, 16, 17 and 18 (English verse-by-verse and prose renderings, valmikiramayan.net)