The phrase is In Persona Christi Capitis, and the Catholic belief is that the priest actually becomes Christ.
6. Just as Christ was sent by the Father, so also He sent the apostles, filled with the Holy Spirit. This He did that, by preaching the gospel to every creature (14), they might proclaim that the Son of God, by His death and resurrection, had freed us from the power of Satan (15) and from death, and brought us into the kingdom of His Father. His purpose also was that they might accomplish the work of salvation which they had proclaimed, by means of sacrifice and sacraments, around which the entire liturgical life revolves. Thus by baptism men are plunged into the paschal mystery of Christ: they die with Him, are buried with Him, and rise with Him (16); they receive the spirit of adoption as sons “in which we cry: Abba, Father” ( Rom. 8 :15), and thus become true adorers whom the Father seeks (17). In like manner, as often as they eat the supper of the Lord they proclaim the death of the Lord until He comes (18). For that reason, on the very day of Pentecost, when the Church appeared before the world, “those who received the word” of Peter “were baptized.” And “they continued steadfastly in the teaching of the apostles and in the communion of the breaking of bread and in prayers . . . praising God and being in favor with all the people” (Acts 2:41-47). From that time onwards the Church has never failed to come together to celebrate the paschal mystery: reading those things “which were in all the scriptures concerning him” (Luke 24:27), celebrating the eucharist in which “the victory and triumph of his death are again made present” (19), and at the same time giving thanks “to God for his unspeakable gift” (2 Cor. 9:15) in Christ Jesus, “in praise of his glory” (Eph. 1:12), through the power of the Holy Spirit.
7. To accomplish so great a work, Christ is always present in His Church, especially in her liturgical celebrations. He is present in the sacrifice of the Mass, not only in the person of His minister, “the same now offering, through the ministry of priests, who formerly offered himself on the cross” (20), but especially under the eucharistic species. By His power He is present in the sacraments, so that when a man baptizes it is really Christ Himself who baptizes (21). He is present in His word, since it is He Himself who speaks when the holy scriptures are read in the Church. He is present, lastly, when the Church prays and sings, for He promised: “Where two or three are gathered together in my name, there am I in the midst of them” (Matt. 18:20) .
Christ indeed always associates the Church with Himself in this great work wherein God is perfectly glorified and men are sanctified. The Church is His beloved Bride who calls to her Lord, and through Him offers worship to the Eternal Father.
Rightly, then, the liturgy is considered as an exercise of the priestly office of Jesus Christ. In the liturgy the sanctification of the man is signified by signs perceptible to the senses, and is effected in a way which corresponds with each of these signs; in the liturgy the whole public worship is performed by the Mystical Body of Jesus Christ, that is, by the Head and His members.
From this it follows that every liturgical celebration, because it is an action of Christ the priest and of His Body which .s the Church, is a sacred action surpassing all others; no other action of the Church can equal its efficacy by the same title and to the same degree.
- From Sancrosanctum Concillium (II Vatican)
The office of priests, since it is connected with the episcopal order, also, in its own degree, shares the authority by which Christ builds up, sanctifies and rules his Body. Wherefore the priesthood, while indeed it presupposes the sacraments of Christian initiation, is conferred by that special sacrament; through it priests, by the anointing of the Holy Spirit, are signed with a special character and are conformed to Christ the Priest in such a way that they can act in the person of Christ the Head.(10)
- From the Decree Prebyterorum Ordinis (II Vatican)
Hope the above clears it up a bit
From the Catachism-
“-1545 The redemptive sacrifice of Christ is unique, accomplished once for all; yet it is made present in the Eucharistic sacrifice of the Church. The same is true of the one priesthood of Christ; it is made present through the ministerial priesthood without diminishing the uniqueness of Christ’s priesthood: “Only Christ is the true priest, the others being only his ministers.”19”
“1548 In the ecclesial service of the ordained minister, it is Christ himself who is present to his Church as Head of his Body, Shepherd of his flock, high priest of the redemptive sacrifice, Teacher of Truth. This is what the Church means by saying that the priest, by virtue of the sacrament of Holy Orders, acts in persona Christi Capitis:23”
The key word in 1548 is “ACTS”
If that's what their Church teaches, they have a real problem, cuz none of their flock believe 'em. ;o)
Why don’t we go out to the Catechism?
874 Christ is himself the source of ministry in the Church. He instituted the Church. He gave her authority and mission, orientation and goal:
In order to shepherd the People of God and to increase its numbers without cease, Christ the Lord set up in his Church a variety of offices which aim at the good of the whole body. The holders of office, who are invested with a sacred power, are, in fact, dedicated to promoting the interests of their brethren, so that all who belong to the People of God . . . may attain to salvation.
875 “How are they to believe in him of whom they have never heard? And how are they to hear without a preacher? And how can men preach unless they are sent?” No one - no individual and no community - can proclaim the Gospel to himself: “Faith comes from what is heard.” No one can give himself the mandate and the mission to proclaim the Gospel. The one sent by the Lord does not speak and act on his own authority, but by virtue of Christ’s authority; not as a member of the community, but speaking to it in the name of Christ. No one can bestow grace on himself; it must be given and offered. This fact presupposes ministers of grace, authorized and empowered by Christ. From him, bishops and priests receive the mission and faculty (”the sacred power”) to act in persona Christi Capitis; deacons receive the strength to serve the people of God in the diaconia of the liturgy, word and charity, in communion with the bishop and his presbyterate. The ministry in which Christ’s emissaries do and give by God’s grace what they cannot do and give by their own powers, is called a “sacrament” by the Church’s tradition. Indeed, the ministry of the Church is conferred by a special sacrament.
876 Intrinsically linked to the sacramental nature of ecclesial ministry is its character as service. Entirely dependent on Christ who gives mission and authority, ministers are truly “slaves of Christ,” in the image of him who freely took “the form of a slave” for us. Because the word and grace of which they are ministers are not their own, but are given to them by Christ for the sake of others, they must freely become the slaves of all.
Rather than ascribing practices to the One Holy Catholic and Apostolic Church, you might simply go out to the Catechism or other official Catholic site and see what actually is.