Dom Mark Kirby OSB states:
A priest is a man set ablaze with a heavenly fire. He cannot approach the altar to offer the Holy Sacrifice without becoming part of the holocaust. When a priest offers himself with the Lamb as a whole burnt offering made over to God utterly and irrevocably, the fragrance of his personal oblation mingles with the sweet fragrance of Christ, and so finds acceptance in the Father’s sight. The fruitfulness of a priest’s intercession is proportionate to his victimhood.
With the Divine Victim
As priest, one stands before the altar to make the sacrificial offering; as victim, one lies upon the altar to be offered in sacrifice. The victimhood of the priest is an exemplary victimhood: it constitutes the single most compelling invitation for the people to participate in the Holy Sacrifice of the Mass by offering themselves together with the Divine Victim. This is the clear teaching of Pope Pius XII in Mediator Dei:
In order that the oblation by which the faithful offer the divine Victim in this sacrifice to the heavenly Father may have its full effect, it is necessary that the people add something else, namely, the offering of themselves as a victim. (Mediator Dei, art. 98)
The priest approaches the altar to offer the Holy Sacrifice to God. The faithful, standing behind the priest, adhere to his mediatorship and, so, are drawn after him into the heavenly fire of the Sacrifice. Even in the absence of the faithful,the priest represents them, intercedes for them, and brings their adoration, thanksgiving, expiation, and petition before God, offering the Holy Sacrifice for the living and the dead.