Romans 16:17-18 meaning

Throughout this entire letter, Paul has been arguing against the competing Jewish "authorities" who desire to overthrow grace and elevate religious rules. He has been reminding the believers in Rome of the true gospel. He is dealing with slanderous statements made in Rome that have misrepresented the gospel he preaches (Romans 3:8). Here, Paul gives one last warning to the believers, desiring that they not be led astray: Now I urge you, brethren, keep your eye on those who cause dissensions and hindrances contrary to the teaching which you learned, and turn away from them (v 17). 

It is also likely that Paul is telling the believers in Rome "Listen to the people I just listed and follow them, but do not listen or follow those they are contesting." One of the pictures of righteousness Paul emphasizes in this letter is unity. Like the body we are to be one. But when it comes to fundamental truth about the gospel, sometimes we have to pick sides. And Paul is making it clear how to choose— choose to side with what is true.

Paul explains how to deal with brothers in error in 1 and 2 Corinthians. By refusing to enable their error, and even refusing fellowship with them in their sin, we are actually helping them. What we want people to do is follow the truth. We help no one by enabling them to continue in error. The goal is not to hate them, but to restore them; if they repent, they should be received gladly back into fellowship (2 Corinthians 2:7-8).

Paul is warning more broadly about false teachers, whether that be the Jewish "authorities" or anyone else. Some people did and will seek to hinder the work of the gospel. Paul has been reminding the believers of their freedom in Christ and that they now have the ability to obey Him daily. 

The Jewish "authorities" were claiming that you had to follow the law (their religious rules) to be justified before God. This argument is opposed to the grace available in Christ and is contrary to the gospel Paul taught (Romans 3:24). Paul wants the believers to be vigilant and to avoid being led astray by false teachers, and he asks them to turn away or keep away from those who preach anything contrary to the gospel. 

In general, we as believers are told to have a very high tolerance level for others—except for false teachers. That is a cancer to the body of Christ that must be resisted and defeated.

Then, Paul describes those who preach contrary to the gospel: For such men are slaves, not of our Lord Christ but of their own appetites; and by their smooth and flattering speech they deceive the hearts of the unsuspecting (v 18). 

He describes them as being enslaved to their own appetites. Although they are slaves, they are very good at making their lies sound good. They use persuasive speech and are often pleasing to listen to. Paul says they deceive those who are unsuspecting, reminding the believers to be watchful for false teachers. These false teachers are to be identified and resisted. It is false teachers to whom we are to apply Jesus's admonition "You will know them by their fruits" (Matthew 7:15-16).