These verses come immediately after Jesus' teaching about the role of the apostles, and then the church, in leading errant believers to repentance (Matthew 18:15-17).  The step-by-step process to reconciliation is firstly by gracious private conversation (Galatians 6:1-3), then by the added testimony of others and, if there is no repentance, the problem should be exposed to the church and the sinner excluded from fellowship.  The style of language is of a court making a judgement.  ‘Binding’ and ‘loosing’ by a prayerful church have divine authority as in Matthew 16:19.   

However, these verses have been used by some, out of context, as a ‘spiritual formula’, to demand for God to give them anything they want.  It is certainly true that God's people, and their agreement in prayer, do make a real difference as God directs (Philippians 1:19; Philemon 1:22).  Jesus promised that when believers agree together with the Lord and pray, He will take action and answer prayer (John 14:13-14; John 15:7-8).  But these verses are in the context of dealing with persistent sin within the church.  

In the face of such a gross issue (which should be rare), the instruction is to come together to pray, seeking God’s discernment about the errant believer.  As the Holy Spirit moves their hearts, they work with God, declaring the truth to the church and calling the person to repentance.  The goal is that sinful believers may repent and be restored (James 5:19-20).  If they are initially unwilling to repent, those believers should be excluded from the security of fellowship and can no longer expect to be comforted in the church (1 Corinthians 5:3-5).  The Apostle Paul wrote that the hope is to induce a ‘godly sorrow that leads to repentance’ (2 Corinthians 7:10-11), followed by restoration to the fellowship in love and full affirmation of God’s forgiveness (2 Corinthians 2:5-8). 
 
The attitude, so common in Western churches, that the church has no right to interfere with the way a dissident believer lives, is not supported by these verses and the rest of the chapter.  Indeed, if the church is not concerned that the Name of the Lord is being slandered by the ungodly behaviour of a believer, it is spiritually sick.  The Lord delegates a measure of accountability for the spiritual health of the church, with the responsibility to pray and act about those matters which diminish the gospel, degrade the church and detract from God’s glory.  But when God's people pray and declare the truth, He works powerfully in His people: they can be restored, and He is glorified.