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The answer can be simply stated: original sin is propagated from parent to child through sexual intercourse.
But to fully appreciate it, I recommend reading a good theological treatise on St. Augustine's doctrine of the Original Sin to understand how the transmission relates to other aspects of St. Augustine's doctrine of the original sin, and what motivated him to construe it that way.
An example of such a treatise from the Catholic perspective can be found in Villanova University Professor Jesse Couenhoven's 2010 journal article published in Augustinian Studies: St. Augustine's Doctrine of Original Sin (pdf here) in which he discusses the following 5 tenets of the doctrine:
In brief, the five elements of the doctrine of original sin are as follows: (1) the source of original sin is a primal sin in the garden of Eden. (2) All human beings share in this sin because of our solidarity with Adam, the progenitor of the race. The results of the primal sin are twofold. (3) From birth, all human beings have an inherited sin (original sin itself), which comes in two forms: common guilt, and a constitutional fault of disordered desire and ignorance. (4) In addition, Augustine holds that the human race suffers a penalty because of sinβhuman powers are weakened, and we will experience death. (5) Finally, Augustine speculates about how both sin and penalty are transmitted from generation to generation. Each of these points are explored below, though I focus on the third, the conceptual heart of the doctrine.
Below are some topic headings as I skimmed through the article quickly.
Primal sin: source of original sin (peccatum originis originans), to be distinguished from condition of original sin (peccatum originatum)
Solidarity with Adam: how Augustine explains the connection with our sin with Adam's sin
Sin as concupiscence (i.e. perversity and lack of order, a turning away from the Creator who is more excellent toward creatures which are inferior to Him), often misunderstood as blaming the body for sin because of Augustine's use of the descriptor "carnal". Under this heading, the role of disordered sexual desire.
Penalty of primal sin arising from "inherited sin" vs. penalty for our own sins
Transmission of original sin
we were in Adam because we were in his seed. Sex matters because that is how human nature is propagated.
carnal concupiscence, though originally the daughter of sin, is also the mother of sin ... lust becomes causally involved in the transmission of original sin ... Sexual lust thereby becomes not merely a symbol of carnal concupiscence, but its cause.
Augustine further claims that parental sins can increase the original sin of their children, and parental righteousness can lessen it. ... Adam's sin had singular power, guaranteeing the presence of original sin in all infants. But parents, too ... ensnare children in their own guilt.