That mistake is a distortion of the people's response to the priest's exhortation to "pray brethren" (Orate fratres).
Part One: priest and people offer their sacrifice of praise and thanksgiving which is symbolized by the gifts of bread and wine.
Orate, fratres, ut meum ac vestrum sacrificium acceptabile fiat apud Deum Patrem omnipotentem.Suscipiat Dominus sacrificium de manibus tuis, ad laudem et gloriam nominis sui, ad utilitatem quoque nostram, totiusque Ecclesiae suae sanctae.
The response of the people (...) makes clear that (the sacrifice) is offered at the hands of the priest who stands in the person of Christ. This prayer, prayed as we move toward the Eucharistic Prayer, reinforces the truth that the sacrifice is offered to the praise and glory of the Father, for the salvation of all present and for the entire holy Church. The new translation also includes, as a modifier of Church, the word “holy” which, while present in the Latin, was omitted from the previous English translation.—Archdiocese of New York
The new translation of the prayer reads (rubrics in RED):
Standing at the middle of the altar, facing the people, extending and then joining his hands, he says:
Pray, brethren (brothers and sisters), that my sacrifice and yours may be acceptable to God, the almighty Father.
The people rise and reply:
May the Lord accept the sacrifice at your hands for the praise and glory of his name, for our good and the good of all his holy Church.
The Priest kisses the altar and, turning towards the People, extending and then joining his hands, says aloud:
Pray, brethren, that my sacrifice and yours may be acceptable unto God, the Father Almighty.
The People stand and respond:
May the Lord accept the sacrifice at thy hands, for the praise and glory of his Name, for our good and the good of all his holy Church.
Too often we hear the people respond with "this sacrifice", which misses not only the unity of priest's and people's sacrifice, a bias is created against the connection of the Offertory to the one and same Sacrifice of Calvary that is re-presented at or on the altar at the hands of the priest. Jesus Christ, through His priest, makes present His Sacrifice upon Calvary for the salvation of man.
Part Two: the offerings of bread and wine, symbolizing our very selves, our offer of praise and thanksgiving to the Father, are transformed by the Holy Spirit into the living Sacrifice, the very Body and Blood of Jesus Christ Whose Sacrifice is the perfect prayer to the Father (cf Catholic Catechism 1322 - 1419)
The Eucharist is a true sacrifice, not just a commemorative meal, as "Bible Christians" insist. The first Christians knew that it was a sacrifice and proclaimed this in their writings. They recognized the sacrificial character of Jesus’ instruction, "Do this in remembrance of me" (Touto poieite tan eman anamnasin; Luke 22:19, 1 Cor. 11:24–25) which is better translated "Offer this as my memorial offering."
Thus, Protestant early Church historian J. N. D. Kelly writes that in the early Church "the Eucharist was regarded as the distinctively Christian sacrifice. . . . Malachi’s prediction (1:10–11) that the Lord would reject Jewish sacrifices and instead would have "a pure offering" made to him by the Gentiles in every place was seized upon by Christians as a prophecy of the Eucharist. The Didache indeed actually applies the term thusia, or sacrifice, to the Eucharist. . . .
"It was natural for early Christians to think of the Eucharist as a sacrifice. The fulfillment of prophecy demanded a solemn Christian offering, and the rite itself was wrapped in the sacrificial atmosphere with which our Lord invested the Last Supper. The words of institution, ‘Do this’ (touto poieite), must have been charged with sacrificial overtones for second-century ears; Justin at any rate understood them to mean, ‘Offer this.’ . . . The bread and wine, moreover, are offered ‘for a memorial (eis anamnasin) of the Passion,’ a phrase which in view of his identification of them with the Lord’s body and blood implies much more than an act of purely spiritual recollection" (J. N. D. Kelly, Early Christian Doctrines [Full Reference], 196–7).—Catholic Answers
(T)he rubrics of the (Ordinary Form) Missal and the directions of the GIRM many times presuppose that the new Mass would be celebrated ad orientem. Facing with the people is not something which belongs only to the Traditional Latin Mass, it is also a legitimate part of the Mass of the Second Vatican Council. In no way should ad orientem be thought of as “going back” or “pre-Vatican II” – this is simply the normal way that the Church presupposed the new Mass would be celebrated, as is indicated by the rubrics and directives in the GIRM.
Interpreting the directions of the GIRMThe GIRM does not always indicate which way the priest should be facing. The clearest indications are found at the “Orate fratres” (the “pray, brethren” before the prayer over the gifts) and at the Communion Rite. In these places, the priest is specifically directed to turn and face the people at certain moments, and then to turn to face the altar at others. We also note that, in the opening rites of the Mass, the priest is directed to face the people for the Sign of the Cross and the greeting, but nothing further is said.Following the Church's tradition and her perennial liturgical practice, it would be safe to assume that the priest is always meant to be facing towards the altar “ad orientem” excepting for those places in which he is expressly directed to face the people. This is the most natural way to read the GIRM, since the indication to face the people is given as a change in direction -- the priest had been facing towards the altar, towards the front of the church, UNTIL he was directed to turn and face towards the people.Consider two examples to illustrate this point:1) In the entrance rites, the procession to the altar has directed the priest in a forward movement -- he clearly approaches the sanctuary facing forward, towards the altar and towards the front of the church. The priest only faces towards the people at that moment when the GIRM directs a change in momentum as he faces the people to make the sign of the Cross. In the entrance rites, the priest is facing ad orientem, except when the GIRM indicates otherwise.2) At the Offertory it is presumed by the GIRM that the priest is facing ad orientem until he is directed to turn and face the people as he says, “Orate fratres” ("pray, brethren”). He then returns to facing toward the altar and towards the front of the church for the Eucharistic prayer.When the GIRM and the Missal give no clear indication of which way the priest is to face, it is presumed that he should be facing ad orientem with the people towards the Lord.