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?
I think you misunderstood the role of Baptism in Catholicism.
Cathecism of the Catholic Church-1263 says
 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.
Not being a Roman Catholic, I cannot give you an answer from the RC perspective and tradition. I can, however, give you a brief overview about what the Bible has to say about Christian baptism.
As a wise man once told me,
"Baptism doesn't make you a Christian; it marks you as a Christian."
As good as infant baptism may be as a way for Christian parents to "sanctify" their offspring through a well-established rite (which exists in both the Catholic and Protestant traditions), nowhere in the New Testament is there any indication that infant baptism cleanses the infant from "original sin." If the RC church teaches this, all I can say is, the teaching is not biblical.
On the basis of Scripture, I say confidently but lovingly, the physical act of being baptized neither forgives us of sin nor frees us from sin. To say otherwise is to contradict the clear teaching of Scripture. If, however, Scripture is not the ultimate guide for your life and practice, then none of the following will be of help to you, and will certainly not answer your question--though I am glad you asked it!
There are two baptisms in the New Testament, each having two aspects, both spiritual and literal/physical. The first is the baptism of John.
John the Baptizer fulfilled two roles: the first as a herald for the Messiah, and the second as a forerunner for the Messiah.
(1) John as a herald. In John's day, just as the king's servant who preceded his king during royal tours from city to city would announce in a loud voice, "The king is coming, the king is coming, get ready for the coming of your king," so too did John announce the coming of the King Jesus, the Messiah.
John, however, shouted
"Repent, the kingdom of heaven is at hand, the kingdom of heaven is at hand, the kingdom of heaven is at hand" (Matthew 3:2).
John certainly attracted the attention of the people with his announcement of the coming kingdom, and they came to him in great numbers.
(2) John as a forerunner. Historically, when a king was scheduled to enter a city, he was not only preceded by a herald, but he also sent servants called forerunners ahead of him to remove any obstacles such as rocks, tree branches, or potholes--anything that would impede his path as he was being carried by litter into the city.
What was hindering the entrance of Jesus the Messiah into the hearts of the people? Sin. John therefore had to tell the people
"'Make ready the way of the LORD, Make His paths straight'" (Matthew 3:3; Isaiah 40:3).
Just as the forerunners of old were responsible to remove obstacles from the road, so too was John responsible to get people to repent of their sins, removing anything that would hinder the entrance of Messiah into their hearts and lives.
John's message to the crowds who came to hear him was a simple one:
And repent the crowds did! Repentance is simply a change of mind resulting in changed behavior. It's doing an "about face": you are going in one direction, but you make a 180-degree turn and go in the opposite direction.
" . . . and [the people from Jerusalem and the surrounding areas] were being baptized by him in the Jordan River, as they confessed their sins" (Matthew 3:2,6).
Before a watching world, the people baptized by John were declaring openly in the waters of the Jordan River--that's the literal/physical part of baptism--what had transpired inside their hearts--that's the spiritual part of baptism; namely, that they were turning their backs on their sins in preparation for the appearance of the coming Messiah, whose coming John foretold.
Being baptized did not cleanse anyone of sin, it merely bore witness to their repentance. On the other hand, the baptism of the Messiah was completely different. As John himself said of Jesus, the coming Messiah and king:
". . . this is the One who baptizes in the Holy Spirit" (John 1:33).
The second baptism, then, is Christian baptism. Like John's baptism, it has both a literal/physical aspect and a spiritual aspect.
The physical aspect is the outward act of being immersed in (or sprinkled by) actual water in the presence of witnesses, and
". . . in the name of the Father and the Son and the Holy Spirit" (Matthew 28:19).
The spiritual aspect is the inward process of conversion by the Holy Spirit, which takes place before we are baptized. Call it regeneration, as in
". . . the washing of regeneration and renewing by the Holy Spirit" (Titus 3:5),
or call it being born again or born from above, as in Jesus' words,
"Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God. . . . [and] unless one is born of water and the Spirit he cannot enter into the kingdom of God" (John 3:3,5).
Christian baptism is for believers in Jesus Christ who have been regenerated within their spirits by the Spirit of Christ, whom Jesus called a "Helper," "the Spirit of truth" who "teaches us all things," and the presence of Christ within us "forever" (see John 14:16 and ff.). Believers who are indwelt by the Holy Spirit want to bear witness to that transformation by obeying the command of our Lord to be baptized. To this day, in some countries of the world, being baptized openly and publicly is to invite persecution and even death!
As Paul tells us in Romans 6, Christian baptism is a symbol of our having died to our old way of life. When we die, we need to be buried. Baptism, then, is a picture of burial that is enacted symbolically by immersing ("burying") a person in water. As the person comes up out of the water, symbolically he is coming alive in the Spirit of God, just as Jesus came alive in His glorious resurrection from the dead (vv.3-5 and ff.).
Does being baptized mean we will never sin again? No. Neither an infant who has been baptized nor a person who has experienced the new birth and has been baptized will cease to sin (see 1 John 1:8-10). We can, however, have victory over sin in our lives, since as believers in Jesus
". . . our old self was crucified with Him, in order that our body of sin might be done away with, so that we would no longer be slaves to sin" (Romans 6:6).
Furthermore, we have the responsibility as baptized believers (literally and spiritually) to
". . . consider [ourselves] to be dead to sin, but alive to God in Christ Jesus";
"not [to] let sin reign in [our] mortal body so that [we] obey its lusts";
"[to] present [ourselves] to God as those alive from the dead, and [our] members as instruments of righteousness to God";
"[because] sin shall not be master over [us], for [we] are not under law but under grace" (Romans 6:11-14).
In conclusion, being baptized with water neither cleanses us from sin nor takes away our ability to sin. It is, however, a way of telling both God and man that we have become new creatures in Christ, that
"the old things passed away; behold, new things have come" (2 Corinthians 5:17).