What does Romans 15:14 mean?

This verse begins a new section of Paul's letter to the Christians in Rome. It's likley he has not yet met most of his original readers in person. He does know Priscilla and Aquila, who host a house church in Rome (Romans 16:3–5). He knows other people in Rome, as well (Romans 16:5–15), so he likely has received several reports about what's going on among the believers in that city.

Based on this knowledge, Paul can both praise and express his concern for them. Here, he calls them brothers and sisters and tells them that he is satisfied in two things about them. First, he believes the Roman Christians to be full of goodness, meaning they have a reputation for doing the right things and being kind and generous.

Second, Paul understands them to have good knowledge and to have the ability to teach that knowledge to each other. By this, Paul means that he believes they have learned well from their teachers about Christian truths to the point that many of them are even able to teach those truths to each other.

Likely, these two ideas are connected. Paul believes the Romans Christians to be full of goodness in their behavior because they are full of knowledge about God and His grace to them in Christ.