The Apostleship of Prayer, with its spirituality of "offering it up," is a Eucharistic spirituality. It's a way to fulfill our responsibilities as the baptized faithful who share in the priesthood of Jesus Christ. We call this the "Priesthood of the Faithful." As "priests" in this sense, all the baptized are called to offer worship to God. The worship that we offer is the sacrifice of our lives, which St. Paul called "a living sacrifice" (see Romans 12: 1).
In the second reading from the Church's Office of Readings today, there is a quote from St. Augustine's autobiographical "Confessions." It goes like this:
Good Father, how you loved us, sparing not your only Son but delivering him up for us sinners! How you loved us, for whose sake he, thinking it no robbery to be equal with you, was made subject to death on the cross. He alone, free among the dead, had the power to lay down his life and the power to take it up again. For our sake he became in your sight both victor and victim--victor, indeed, because he was victim. For our sake, too, he became before you both priest and sacrifice--priest, indeed, because he was a sacrifice....
Baptized into the Body of Christ, all the faithful are called to be sacrificial victims and victorious priests. We are called, as members of Christ's Body, to join the offering of ourselves and our day to the perfect offering of the Head of the Body, Christ. This is what living the Eucharist means. Making this offering unites us to the one who was victorious over sin and death so that we too will pass from this life victorious.