As was said in Part 2, the Sacrament of Holy Orders (ordination) causes a real change in the very essence of the one receiving the Sacrament. From the Catechism of the Catholic Church:
This sacrament configures the recipient to Christ by a special grace of the Holy Spirit, so that he may serve as Christ's instrument for his Church. By ordination one is enabled to act as a representative of Christ, Head of the Church, in his triple office of priest, prophet, and king.
...The sacrament of Holy Orders...confers an indelible spiritual character and cannot be repeated or conferred temporarily.
...he cannot become a layman again in the strict sense, because the character imprinted by ordination is for ever. The vocation and mission received on the day of his ordination mark him permanently. (Catechism, 1581-83)
Keep in mind then that ordination literally changes the person in a permanent way. Being a priest is not just about public preaching or spiritual counseling or managing a parish. While these practical things do matter, the priesthood is ultimately about being configured to the priesthood of Christ. With Jesus as the model of perfect priesthood, (He is the High Priest and the source of our earthly priesthood), the recipient of ordination is changed to conform to this ideal of priesthood that we see in Jesus. This transformation is so profound that the priest is said to stand in persona Christi (in the person of Christ) when he is exercising his priestly role.
Who can properly receive such a transformation? Who can stand in the person of Christ? As with any Sacrament we must ask, what requirements must be met for the Sacrament to be valid?
In the Sacrament of the Eucharist, we know that Jesus took bread, gave thanks, broke it, and gave it to His disciples. In order for our Eucharist to be valid we must also use bread. Bread is the required stuff for the Sacrament. The same is true of the wine. To remain faithful to Jesus' own actions, bread and wine must be used in order for the Eucharist to do what we say it does - become the very Body and Blood of Jesus.
So too in ordination. In order for a valid ordination to take place, the right stuff must be present. We know that the priest will be standing in persona Christi - in the person of Christ - and as a person we know that Jesus was a man. So it is logical that a man must be the correct recipient for a valid ordination to take place.
Some might argue that men and women are no different when it comes to spiritual matters. God is neither male nor female, He is pure spirit, and so when it comes to the spiritual care of the Church our sex should not matter. But this is contrary to the Catholic Church's teaching on human spirituality. We are never merely "spiritual" beings if we are truly human. The Church teaches that human beings possess both a soul and a body, and these two realities are intimately united:
The unity of soul and body is so profound that one has to consider the soul to be the "form" of the body: i.e., it is because of its spiritual soul that the body made of matter becomes a living, human body; spirit and matter, in man, are not two natures united, but rather their union forms a single nature. (Catechism 365)
So when we say that Jesus was a man, that the Son of God came down to earth and became a man, we mean that He was a man in the same sense that every man is. His body and soul, intimately united, in a male human form. When a priest stands in persona Christi, he stands in the person of a man named Jesus of Nazareth. Jesus is not a pure spirit; He is a male human composed of both body and soul.