Question: According to Catholic thought, if we are washed of Original Sin at the time of baptism, why do we transmit Original Sin to our children? In other words, how is it that my child is born with Original Sin when I was freed from Original Sin at the time of my baptism?

Answer: 
I think you misunderstood the role of Baptism in Catholicism. 

Cathecism of the Catholic Church-1263 says

> 
> [1263] By Baptism all sins are **forgiven**, original sin and all personal
>  sins, as well as all punishment for sin. In those who have been
>  reborn nothing remains that would impede their entry into the Kingdom
>  of God, neither Adam's sin, nor personal sin, nor the consequences of
>  sin, the gravest of which is separation from God.
> 
> 
> 

It doesn't say original sin is washed away, it says it is forgiven. Forgiven doesn't mean you don't have original sin anymore; it only means your sins are forgiven.

Our sins cannot be washed away by merely washing with water nor any kind of Baptism. Only the blood of Jesus Christ can wash away our sins.

> 
> *1 John 1:7 (NIV)* But if we walk in the light, as he is in the light,
>  we have fellowship with one another, and the **blood of Jesus**, his Son,
>  **purifies us from all sin**.
> 
> 
> *Revelation 7:14 (NKJV)* And I said to him, “Sir, you know.” So he said
>  to me, “These are the ones who come out of the great tribulation, and
>  **washed their robes** and made them white **in the blood of the Lamb**.
> 
> 
> *Revelation 22:14 (DRA)* Blessed are they that **wash their robes in the
>  blood of the Lamb**: that they may have a right to the tree of life, and
>  may enter in by the gates into the city.
> 
> 
>
Original sin is a state of human nature passed to all mankind, but it is not like a gene passed on from our immediate parents. It affects the nature of all humans. The Catholic Catechism discusses this in and around paragraphs 396 to 408.

In paragraph 404 talks a bit about what Original Sin is and its source:

> 
> 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 [Adam and Eve] would then transmit in a fallen state**. 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.
> 
> 
> 

In paragraph 405 talks about what Original Sin is and the effects of Baptism on it:

> 
> Although it is proper to each individual, original sin does not have the character of a personal fault in any of Adam's descendants. It is a **deprivation of original holiness and justice**, but human nature has not been totally corrupted: it is wounded in the natural powers proper to it, subject to ignorance, suffering and the dominion of death, and inclined to sin — **an inclination to evil that is called concupiscence**". Baptism, by imparting the life of Christ's grace, **erases original sin and turns a man back towards God, but the consequences for nature, weakened and inclined to evil, persist in man and summon him to spiritual battle**.
> 
> 
> 

On the effects of Baptism, paragraph 978 says:

> 
> "When we made our first profession of faith while receiving the holy Baptism that cleansed us, the forgiveness we received then was so full and complete that there remained in us absolutely nothing left to efface, neither original sin nor offenses committed by our own will, nor was there left any penalty to suffer in order to expiate them. ... **Yet the grace of Baptism delivers no one from all the weakness of nature**. On the contrary, we must still combat the movements of concupiscence that never cease leading us into evil "
> 
> 
> 

Finally, in talking about Baptism of Infants in paragraph 1250:

> 
> Born with a fallen human nature and tainted by original sin, children also have need of the new birth in Baptism to be freed from the power of darkness and brought into the realm of the freedom of the children of God, to which all men are called. The sheer gratuitousness of the grace of salvation is particularly manifest in infant Baptism. The Church and the parents would deny a child the priceless grace of becoming a child of God were they not to confer Baptism shortly after birth.
> 
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