What is happening here
Every Catholic Mass begins the same way: the priest, with the assembly, makes the Sign of the Cross — "In the name of the Father, and of the Son, and of the Holy Spirit. Amen." It is the simplest and oldest of Christian gestures, and it is doing serious work. By tracing the cross on our own bodies, we are placing what we are about to do under the sign of Christ's death and resurrection. By naming the Persons of the Trinity, we are making explicit whose company we have entered.
The greeting that follows — most often "The Lord be with you," with the assembly responding "And with your spirit" — is not a casual hello. It is an ancient liturgical formula that appears in the Old Testament (the angel's greeting to Gideon, Boaz's greeting to his harvesters) and is used by Paul throughout his letters. The "with your spirit" response is a reference to the Holy Spirit who has been given to the priest in ordination. The dialogue is the Church recognizing, at the start of the Mass, that the Lord is present here in a particular way through his ordained minister.
For Catholics, this opening is the threshold of the liturgy. Everything that follows — the readings, the homily, the offering, the Eucharist — happens inside the space that this Sign of the Cross and greeting open up.