After the marriage, the wife has to live at her husband's home. What do our scriptures say about this custom?
A girl is wed to someone in order to help him propagate his vamsha and thus continue the gotra which is established by the ancient rishis. Also it enables one to perform the core vedic rituals like Agnihotra and various sacrifices for which spouse is mandatory.
However, in certain circumstances the girl can continue to live with the parent if she has no brother and if she has the responsibility of the parent. Such s girl adopts the duties of a son and hence she is referred to as putrikA. She may be wed to a man who continues to live with her in her house.
She also performs the rituals like shraddha including the sapinda etc for the manes. The progeny born of a putrikA generally belongs to the vamsha of the girl and continues their family lineage.
Putrikādharma (पुत्रिकाधर्म).—A son-less father generally gives his daughter in marriage and takes a promise from the sonin-law that the son of them would be counted as his own. Though Svāyambhuva Manu had a son, he adopted Ākūti's son.*
That girl who has no brother nor father should not be wed, O chief of Bharata's race, for she may be intended as Putrika of her sire. 2 After the appearance of puberty, the girl (if not married) should wait for three years. On the fourth year, she should look for a husband herself (without waiting any longer for her kinsmen to select one for her). The offspring of such a girl do not lose their respectability, nor does union with such a girl become disgraceful. If, instead of selecting a husband for herself, she acts otherwise, she incurs the reproach of Prajapati herself. One should wed that girl who is not a Sapinda of one's mother or of the same Gotra with one's father. Even this is the usage (consistent with the sacred law) which Manu has declared.'
Giving away of the maiden daughter in marriage is called the "KanyA DAnam" or "the gift of maiden daughter". It is the greatest gift a father can make.
And, after a gift has been made the giver loses ownership over it. So, that's why the daughter can not stay any longer with her parents after marriage. She has to live with her husband who's the recipient of the gift here.
For example, the MimAngsA ParibhAshA (MP) defines charity (DAnam) as follows:
sva svatvanivrittipurvakam para svatvApAdanam dAnam ||
Sacrificing ownership over something and assigning ownership of that thing to someone else is dAnam (charity). (MP 56- Karmabheda Nirupanam)
And, that giving away a daughter in marriage is really a charity is established from verses like the ones given below:
No deterioration [of piety] is seen in him who gives away a maiden-daughter [in marriage], and in him who gives life in a disease. These three gifts bear equal fruits.* (332) (Atri Smriti)
Having given away a maiden-daughter, decorated with ornaments, clothes and seats a father attains to the celestial region and is adored of the Suras (celestials . (64) (Samvarta Smriti)
Also as per Manu Smriti, for a wife, serving her husband is equal to staying at her Guru's house, doing the household duties at his residence is equal to the performance of Agnihotra and etc.
VaivAhika vidhi strinAm samskaro vaidikah smritah |
PatisevA gurau vAsoh grihArtho agniparikriyA ||
Manu Smriti 2.67. The nuptial ceremony is stated to be the Vedic sacrament for women (and to be equal to the initiation), serving the husband (equivalent to) the residence in (the house of the) teacher, and the household duties (the same) as the (daily) worship of the sacred fire.
So, that's why the custom is like that.
Moreover, there is yet another rule, mentioned in the Smritis, according to which a father must not partake food in his married daughter's house.
A Kshatriyas food destroys energy, and a S'udra's. food destroys Brahma-vigour. He, who partakes of his daughter's food, eats the dirt of the earth. (298)
If one's own daughter does not give birth to a child, the father should not take his meals at her house. He who takes food out of love, goes to the Puya hell. (299) (Atri Smriti)
So, this rule also requires the daughter not to live with her father after her marriage.