It is supposed that this helps the person with any spiritual conflict that might occur at the moment of death.
The anointing, along with the prayers of the priest, is expected to bring about that person’s salvation as they face death.
Problems with this are many.
First, salvation itself is a work of God and only God. It is not some work that depends on anything that we as human beings do. It is only the work of Christ don already on behalf of believers. Once a person has trusted Jesus Christ as their Savior they are ready to enter the next world. Nothing else is necessary or relevant.
“Context is king!” James did not say, “Is there any one of you dying.” Rather he said, “Is there any one of you sick?” God (through James) is not talking about impending death but rather restoration back to physical (not spiritual) health. To make this passage about death is error.
Finally, it does not say to call for the priest – instead, it says to call for the elders of the church. The leadership of the church is to be called to pray for that person’s health, relief from suffering, and their friends and family who are going through suffering. Remember, each believer in Jesus Christ is a priest.
I recommend the Book of Hebrews, and especially Galatians, chapter 3.
Supporting documentation. There’s much more to say. This is enough to answer your question.
You do not understand Catholic sacramental theology. Catholics have always maintained that sacraments get their effectiveness from the merits of Jesus’s death on the Cross, and nowhere else. The minister contributes nothing but the administration. He is an instrument. The sacraments confer grace (a gift from God), not “salvation” (also a gift from God). You are struggling to impute a fake accusation, works righteousness. The Catholic Church has always taught that salvation comes through faith. That faith also is a gift from God, not a work of the believer, except in so far as the believer willingly assents to the faith given him, cooperating with God in something he could neither have initiated nor completed on his own.