The verse demonstrates several things the Church has taught for 2,000 years. First, in order to confect the sacrament, one must call for the elders or priests of the Church. This requires men specially ordained to do the job, and gets into what we mean by Church (don’t forget about Peter, the keys, dynastic succession, priestly ordination, the power to bind and loose, and the pinnacle and bulwark of the truth). Secondly, James says the priests’ prayer of faith will save the sick man and the Lord will raise him up. This demonstrates that the Church’s priests act in the person of Christ (“in persona Christi”) in furthering Christ’s work of salvation. Yes, Jesus is our only Savior, but He desires us to participate in His eternal priesthood, and He calls certain men to participate in a very intimate way by effecting salvation (through the ministerial priesthood described here). So the priests, through the power of Christ, save the sick man’s soul.
Finally, by virtue of the actions and prayers of the priests, the sick man’s sins are forgiven (this is what actually saves the man’s soul). Protestants have great difficulty with this verse particularly because it demonstrates that priests have the power and authority to forgive sins (which was given to men by Christ; see also Matthew 9:8). Unlike what the Bible provides, no where in its theology or practice does Protestantism provide for priestly forgiveness of sins or the sacrament of the sick.