What does rebirth through the Holy Spirit mean?
According to the doctrine of the Catholic Church rebirth through the Holy Spirit happens during the sacrament of Baptism.
The “Catechism of the Catholic Church” says:
1213 Holy Baptism is the basis of the whole Christian life, the gateway to life in the Spirit (vitae spiritualis ianua), and the door which gives access to the other sacraments. Through Baptism we are freed from sin and reborn as sons of God; we become members of Christ, are incorporated into the Church and made sharers in her mission: ‘Baptism is the sacrament of regeneration through water in the word’. (Cathechismus Romanus 2,2,5)
What is this Sacrament called?
1214 This sacrament is called Baptism, after the central rite by which it is carried out: to baptize (Greek baptizein) means to "plunge" or "immerse"; the "plunge" into the water symbolizes the catechumen's burial into Christ's death, from which he rises up by resurrection with him, as ‘a new creature’. (2 Cor 5:17; Gal 6:15)
1215 This sacrament is also called ‘the washing of regeneration and renewal by the Holy Spirit, (Tit 3:5) for it signifies and actually brings about the birth of water and the Spirit without which no one "can enter the kingdom of God." (jn 3:5)
1216 ‘This bath is called ‘ because those who receive this [catechetical] instruction are enlightened in their understanding . . . .’ (Justin, apol. 1,61,12 9). Having received in Baptism the Word, "the true light that enlightens every man," (Jn 1.9) the person baptized has been "enlightened, (Heb. 10:32) he becomes a "son of light," (Thes 5:5) indeed, he becomes "light" himself: (Eph 5:8).
Baptism is God's most beautiful and magnificent gift....We call it gift, grace, anointing, enlightenment, garment of immortality, bath of rebirth, seal, and most precious gift. It is called gift because it is conferred on those who bring nothing of their own; grace since it is given even to the guilty; Baptism because sin is buried in the water; anointing for it is priestly and royal as are those who are anointed; enlightenment because it radiates light; clothing since it veils our shame; bath because it washes; and seal as it is our guard and the sign of God's Lordship.( St. Gregory of Nazianzus, Oratio 40, 3-4: PG 36, 361C.) See: www.vatican.va/archive/ENG0015/__P3H.HTM