We often think of Baptism as "washing away Original Sin" and Confirmation as a "strengthening" (confirmo in Latin means "I strengthen") in the Church Militant.
Baptism is actually prefigured in the Great Flood and in the Parting of the Sea by Moses. It is the washing away of sin, to be sure, but it is, more importantly, the crossing over into a new and renewed creation, the new promised land and the new Jerusalem which is the Church (and which the old covenant and the old Jerusalem anticipated). We enter the Church in Baptism ("baptism" means "plunging" in Greek, hence when one is "baptized into Christ" one is plunged into the body of Christ, the Church) we enter into something pure and our sins are washed away as a consequence.
Confirmation, or Chrismation, is a similar story. The anointing with oil is actually an Old Testament style anointing to the priesthood. In this case it is the anointing into a priestly group of people (1 Peter 2:9—that passage Luther so abused). In the ancient days the priests came from the Levites. Christ has transformed all members of the Church into members of a new type of people that has this priestly charism, that is, the power and ability to spread the Gospel, which is of course not the same as Holy Orders. It is perhaps a very small participation in Christ's priesthood in spreading the Gospel. The priests and bishops in Holy Orders have a much stronger participation since they have the power to confect the Sacraments. Either way it is a very real participation in the Divine Life. Because it is a joining to a new people, Confirmation was always done in tandem with Baptism: one entered into the new creation and the new people of God at the same time. As infant Baptism (which was always done) became the norm the Byzantines Confirmed infants and the Romans reserved it for the bishop to do. There is, in both the traditional and Novus Ordo Roman books, a remnant of Confirmation of infants: the priest anoints the baptized child with Chrism saying nothing. Thankfully we do confer Confirmation after Baptism with converts now.
Also at Confirmation one receives the Holy Spirit. Because one enters into a priestly people one must receive the Holy Spirit to be guided in truth (1 John 2:20). The traditional Roman rite loses sight of this. The form in the Roman rite is "I sign you with the sign of the Cross and strengthen you with the chrism of salvation in the Name of the Father and of the Son and of the Holy Spirit." The Byzantine rite, which the new Roman books copied in this case, is far clearer "Be sealed with the gift of the Holy Spirit" which really means "Be sealed with the gift that is the Holy Spirit."
Makes me wish I appreciated my own Confirmation more!