There are at least two main ways of looking at them. One way is looking at them in terms of church discipline. Another way of looking at them is in terms of the proclamation of the gospel. But the Roman Catholic apologist's way of looking at the text as supposedly conferring a power of infallibly defining dogma is different from either of those.
At this point, the RC apologist said, "... now we’re down to a matter of interpretation." Yes, it's a matter of interpretation as opposed to say a matter of one person just blatantly saying, "because my church says so." Yet matters of interpretation are often still resolvable based on the text, context, and so forth. So, just labeling something a "matter of interpretation" is not really an out for our friend.
The problem for this particular RC apologist (and others like him) is that his own church, in her official teachings, interprets "binding and loosing" as related to discipline (all quotations from the official Catechism of the Catholic Church):
881 The Lord made Simon alone, whom he named Peter, the "rock" of his Church. He gave him the keys of his Church and instituted him shepherd of the whole flock. "The office of binding and loosing which was given to Peter was also assigned to the college of apostles united to its head." This pastoral office of Peter and the other apostles belongs to the Church's very foundation and is continued by the bishops under the primacy of the Pope.Now, my Roman Catholic friends know very well that disciplinary decisions are not accorded the charism of infallibility in Roman Catholic theology. Yet they argue as though the verse teaches that the power to bind and loose implies infallibility. There's a third way between between their church's view and their view, as mentioned above.
1444 In imparting to his apostles his own power to forgive sins the Lord also gives them the authority to reconcile sinners with the Church. This ecclesial dimension of their task is expressed most notably in Christ's solemn words to Simon Peter: "I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." "The office of binding and loosing which was given to Peter was also assigned to the college of the apostles united to its head."
1445 The words bind and loose mean: whomever you exclude from your communion, will be excluded from communion with God; whomever you receive anew into your communion, God will welcome back into his. Reconciliation with the Church is inseparable from reconciliation with God.
1478 An indulgence is obtained through the Church who, by virtue of the power of binding and loosing granted her by Christ Jesus, intervenes in favor of individual Christians and opens for them the treasury of the merits of Christ and the saints to obtain from the Father of mercies the remission of the temporal punishments due for their sins. Thus the Church does not want simply to come to the aid of these Christians, but also to spur them to works of devotion, penance, and charity.
553 Jesus entrusted a specific authority to Peter: "I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." The "power of the keys" designates authority to govern the house of God, which is the Church. Jesus, the Good Shepherd, confirmed this mandate after his Resurrection: "Feed my sheep." The power to "bind and loose" connotes the authority to absolve sins, to pronounce doctrinal judgements, and to make disciplinary decisions in the Church. Jesus entrusted this authority to the Church through the ministry of the apostles and in particular through the ministry of Peter, the only one to whom he specifically entrusted the keys of the kingdom.
979 In this battle against our inclination towards evil, who could be brave and watchful enough to escape every wound of sin? "If the Church has the power to forgive sins, then Baptism cannot be her only means of using the keys of the Kingdom of heaven received from Jesus Christ. The Church must be able to forgive all penitents their offenses, even if they should sin until the last moment of their lives."
980 It is through the sacrament of Penance that the baptized can be reconciled with God and with the Church:
Penance has rightly been called by the holy Fathers "a laborious kind of baptism." This sacrament of Penance is necessary for salvation for those who have fallen after Baptism, just as Baptism is necessary for salvation for those who have not yet been reborn.981 After his Resurrection, Christ sent his apostles "so that repentance and forgiveness of sins should be preached in his name to all nations." The apostles and their successors carry out this "ministry of reconciliation," not only by announcing to men God's forgiveness merited for us by Christ, and calling them to conversion and faith; but also by communicating to them the forgiveness of sins in Baptism, and reconciling them with God and with the Church through the power of the keys, received from Christ:
[The Church] has received the keys of the Kingdom of heaven so that, in her, sins may be forgiven through Christ's blood and the Holy Spirit's action. In this Church, the soul dead through sin comes back to life in order to live with Christ, whose grace has saved us.982 There is no offense, however serious, that the Church cannot forgive. "There is no one, however wicked and guilty, who may not confidently hope for forgiveness, provided his repentance is honest. Christ who died for all men desires that in his Church the gates of forgiveness should always be open to anyone who turns away from sin.
As William Webster has explained, the keys of the kingdom of heaven were entrusted to the apostles in the form of entrusting them with the gospel message to proclaim. Thus, the binding and loosing there refers to the proclamation of the true gospel.
Isaiah 61:1The Spirit of the Lord God is upon me; because the Lord hath anointed me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound;
Luke 4:18The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised,After all, the gospels are the keys that unlock the gates of hell, allowing the church (all believers - all those who confess that Jesus is the Christ, the son of the Living God) to be resurrected unto eternal life. Those who do not follow the true gospel will not enter into heaven - those who do, will.
The apostles have handed down (traditioned) that gospel to us in the writings of the New Testament. It was once delivered and remains the same today as it was when the apostles delivered it.