What does Romans 15:6 mean?

This verse concludes a prayer for the church in Rome that Paul began in the previous verse. There he asked the God of endurance and encouragement to grant them the ability to live in harmony with each other in Christ.

Now he asks for something that describes the very purpose of the church. We exist, in part, to glorify God the Father of our Lord Jesus Christ. That is what we are here for. One of the ways we accomplish that, according to Paul's prayer, is by living in unity as if singing with one voice. In other words, Paul prays that the Christians in Rome will be unified in their purpose to glorify God together as if they were all singing the same song.

It's a beautiful picture of what life in the church should be like. To get there, though, Paul has made it clear that both the strong and weak in faith (Romans 14:1) will have to yield to each other and refuse to judge each other (Romans 14:13). They will have to set themselves aside to be able to harmonize and sing with one voice.