You are referring to the Fraction Rite. Jesus broke the bread before giving it to the Apostles at the Last Supper.
From the General Instruction of the Roman Missal, section 83:
“The priest breaks the Eucharistic Bread, assisted, if the case calls for it, by the deacon or a concelebrant. Christ’s gesture of breaking bread at the Last Supper, which gave the entire Eucharistic Action its name in apostolic times, signifies that the many faithful are made one body (1 Cor 10:17) by receiving Communion from the one Bread of Life which is Christ, who died and rose for the salvation of the world. The fraction or breaking of bread is begun after the sign of peace and is carried out with proper reverence, though it should not be unnecessarily prolonged, nor should it be accorded undue importance. This rite is reserved to the priest and the deacon. The priest breaks the Bread and puts a piece of the host into the chalice to signify the unity of the Body and Blood of the Lord in the work of salvation, namely, of the living and glorious Body of Jesus Christ.”
Why does the priest say repeatedly “The Lord be with you”?
The ritual statement “The Lord be with you” is an ancient greeting (see Ruth 2:4). It is a blessing, a statement of faith which acknowledges God’s presence, and an expression of desire that each person is seeking to deepen their relationship with God.
This is what the priest says as he breaks the bread:
“Then he takes the host, breaks it over the paten and places a small piece in the chalice, saying quietly: May this mingling of the Body and Blood of our Lord Jesus Christ bring eternal life to us who receive it.” (New Roman Missal)
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