Dear Geoffrey:

The Church teaches this because the Genesis record makes implication to it. The whole point of the Adam and Eve story concerning eating the fruit is offered to explain of how sin came into the world and how each person suffers from it and tends to commit it (actual and habitual concupiscence; see article). The point of Baptism and why it is necessary is to wash away Original Sin (and any personal sins).

I recommend the article on Original Sin from the 1917 Catholic Encyclopedia, and the Catechism beginning at paragraph 396. Especially pertinent to this question is the following:

404 How did the sin of Adam become the sin of all his descendants? The whole human race is in Adam "as one body of one man".293 By this "unity of the human race" all men are implicated in Adam's sin, as all are implicated in Christ's justice. Still, the transmission of original sin is a mystery that we cannot fully understand. But we do know by Revelation that Adam had received original holiness and justice not for himself alone, but for all human nature. By yielding to the tempter, Adam and Eve committed a personal sin, but this sin affected the human nature that they would then transmit in a fallen state.294 It is a sin which will be transmitted by propagation to all mankind, that is, by the transmission of a human nature deprived of original holiness and justice. And that is why original sin is called "sin" only in an analogical sense: it is a sin "contracted" and not "committed" - a state and not an act.

406 The Church's teaching on the transmission of original sin was articulated more precisely in the fifth century, especially under the impulse of St. Augustine's reflections against Pelagianism, and in the sixteenth century, in opposition to the Protestant Reformation. Pelagius held that man could, by the natural power of free will and without the necessary help of God's grace, lead a morally good life; he thus reduced the influence of Adam's fault to bad example. The first Protestant reformers, on the contrary, taught that original sin has radically perverted man and destroyed his freedom; they identified the sin inherited by each man with the tendency to evil (concupiscentia), which would be insurmountable. The Church pronounced on the meaning of the data of Revelation on original sin especially at the second Council of Orange (529)296 and at the Council of Trent (1546).

God Bless,
Bro. Ignatius Mary