Romans 6:17-20 meaning

It's interesting to note that Paul has never met the Roman Christians to whom he's writing. He doesn't know them personally; he didn't plant this church. He says in Chapter 1 that their faith is spoken of throughout the whole world (Romans 1:8). How would Paul know that? He's been throughout the whole world, and he's heard of their faith. He offers thanks to God that though you were slaves of sin, you became obedient from the heart to that form of teaching to which you were committed (v 17). He tells them he is thankful that they have such faith and were delivered into obedience. He tells them he's thankful they're already living obedient lives, and that having been freed from sin, they became slaves of righteousness (v 18). 

The Roman believers addressed in this letter had competing Jewish "authorities" among them trying to discredit Paul's teaching. Paul is exhorting them not to give into the false teaching. His ministry is at risk because his opponents are spreading slander about his teaching in Rome, the most famous city in the known world at that time. It is incredibly important for Paul to clarify his message and reassure the Roman believers that they're already living faithfully; if the competing Jewish "authorities" lead believers in Rome to live under the law, the Gospel of grace Paul has taught throughout the world will be at risk.

In verse 19, Paul clarifies why he is using the term slavery, and it is due to the weakness of man. I am speaking in human terms because of the weakness of your flesh (v 19). Before we were saved, we presented our members as slaves to impurity and to lawlessness. Serving lawlessness and sin only results in further lawlessness (v 19). As weak humans, we will always serve a master, no matter how strong and independent we think we are. Christians who have believed in Jesus now have the power to follow God, and the opportunity to escape sin as a master: Now present your members as slaves to righteousness, resulting in sanctification. For when you were slaves of sin, you were free in regard to righteousness (v 20). Now we can choose righteousness as master. 

However, we are still weak in the flesh, so we must commit ourselves as slaves to righteousness by being obedient to God. This results in sanctification, changing to become more like Christ. Sanctification just means "set apart for a special purpose," and it is the outcome of submitting to God's special design for us. It is living out the resurrection life we have through Christ. This ties back to the theme verses of Romans, in 1:16-17, that the righteous shall live by faith; the person who lives by fully entrusting himself to God will live a harmonious life that God desires for him.

The message of the competing Jewish "authorities" was that you have to obey the law to acquire righteousness, and Paul is attempting to encourage the Christians in Rome not to listen to these false teachers. He is telling them they were already living righteously through faith. Throughout the letter to the Romans, Paul has continuously refuted these contrary teachings, and emphasizes it's all about a person's heart. You can't improve from the outside in, it has to be inside out.

These believers in Rome, whom he's never met personally, are living lives of faithful obedience. Paul is encouraging them in their outward living of their inward faith, charting for them their history of being former slaves to lawlessness, which only created further lawlessness and kept them completely separate from righteousness; but now, through their faith in Christ, and subsequently their faithful obedience in their daily lives, they are being sanctified, and word of their faithfulness has spread all around the world. Paul wants them to continue walking in faith, and not succumb to the false teaching of legalism.