It was a way to secure public recognition of a marriage, as it introduced the new union to the phratry while also signaling a change in status and lifestyle for the groom to the group.
Rather, any children born of a union not between two Athenians could be considered nothoi, meaning bastards, or metics, and excluded from the full rights of citizenship.
If the legitimacy of a son was ever contested on the grounds of parentage, the young man could point to the celebration of his parents' gamelia in the memory of the phratores, signifying they were both true Athenian.
[1] This is seen in the literature of the period, such as in Demosthenes 57, where Euxitheos calls upon phratry members who witnessed his father's gamelia, celebrated on his mother's behalf, to prove his legitimacy.
In regard to the specific practice of gamelia, it appears to have been both an offering of wine and food as a feast for fellow members as well as an accompanying sacrifice.
[1] This suggests that perhaps not all phratries demanded gamelia to follow the marriages of their members, but the benefits made it a popular custom: like the registration of the infant's name, it would be future insurance of the child's political and inheritance rights.
[1] Another unclear aspect of the ritual is whether or not the brides themselves were actually present: given the nature of the male-oriented phratry, they probably would have remained on the margins of the group, rather than playing a central role.
This is possible if the families involved lived close by where the phratry traditionally met, cutting down on travel and festival costs for both parties by hosting it alongside the Apaturia.