Romans 15:14-16 meaning

Paul reassures his audience of Roman believers that he is fully confident that they are living righteously and harmoniously by faith: And concerning you, my brethren, I myself also am convinced that you yourselves are full of goodness, filled with all knowledge and able also to admonish one another (v 14). 

He has not written this letter to instruct them in things they weren't already doing (Romans 1:8). He tells them they are full of goodness. They are also knowledgeable and able to instruct (admonish) each other. The Roman believers are his brethren even though they are Gentiles, because they are all children of God through faith. Paul is convinced of their godly character even though he has not met them (Romans 1:10-13). This is likely because of the reports he has heard of them (Romans 1:8) and the people he knew well, like Priscilla and Aquila, who have probably sent the reports (Romans 16:3). 

Paul does clarify, however, that he knowingly has written certain points to remind them, to edify them in their growth. He says I have written very boldly to you on some points so as to remind you again, because of the grace that was given me from God (v 15). 

This bold writing hearkens back to Paul's rebuke of the competing Jewish "authorities" in Rome, who were slandering his gospel (Romans 3:8). Paul's objectors tried to discredit him (and probably Priscilla and Aquila, who are teaching Paul's gospel in Rome—Romans 16:3; Acts 18:2, 18, 26). The objectors claim that Paul's teaching of the gospel of grace meant believers ought to sin all the time because then God's grace would increase through forgiving us continuously, and that would glorify God by showing His grace. 

Paul did in fact teach that no believer can out-sin God's grace (Romans 5:20). However, Paul has argued vigorously that believers should avoid sin because it brings about a consequence of death, slavery, and loss (Romans 1:24, 26, 28, 6:23; Galatians 5:13, 6:8). 

By framing Paul's teaching this way, the objectors sought to discredit Paul's teaching of grace. These competing Jewish "authorities" argued that the answer was for the Roman believers to return to the Law and obey rules to prove their righteousness. But Paul made clear that this argument made by his opponents contradicts the gospel.

Paul is explicit in that he is only able to write boldly to the Gentile believers because of the grace that was given to him from God (v 15, Romans 3:9). Before Paul became a believer, he had persecuted the church and tried to destroy it (1 Corinthians 15:9). He knows that it is only because of God's grace to him that he was given the mission of taking the gospel and ministering to the Gentiles.

So, throughout this letter, Paul has both discredited their slander and displayed what true righteousness looks like. The Law from the Old Testament never brought righteousness to anyone, only more sin, because rules do not change the heart. The more rules there are, the more opportunities there are to break them. 

Instead, Christ died for us, and by believing in Him we are justified in the presence of God and redeemed from our sin. He gives us the Spirit, which allows us to fulfill the Law when we walk in the Spirit (Romans 8:3). Now, in living out our resurrection life, we are called to live by faith and follow the leading of the Holy Spirit (Romans 8:14). Although it is true that we cannot out-sin the grace of God (Romans 5:20), Paul has made it clear we should avoid sin, and live the resurrected life because it is in our best interest to avoid the negative consequences of sin (Romans 1:24, 26, 28, 6:23).

At the beginning of the letter, Paul thanked God for the famous faith of the Roman believers (Romans 1:8). He knew they were living righteously by faith already, but he also knew their faith was under threat because of the slanderers in Rome (Romans 3:8). So Paul wrote this letter not only to defeat that slander but to remind the Roman believers of the truth of the gospel, which they were already practicing.

The final reason Paul gives for writing this letter is that it's his job. In verse 16, Paul says he is a minister of Christ Jesus to the Gentiles, ministering as a priest the gospel of God, so that my offering of the Gentiles may become acceptable, sanctified by the Holy Spirit (v 16). 

His life purpose is to spread the gospel of grace to the Gentiles and teach them to live righteously through a walk of faith that God's ways are for our best. Jesus appointed Paul to this task, and Paul is doing his best to complete it (Acts 9:1-19). Paul is a minister of Christ Jesus, meaning that he is acting on behalf of Christ. Jesus appointed Paul to minister to the Gentiles and Paul was obedient to His command (Acts 9:15). 

Paul recognized his commission from God, and said "woe is me if I do not preach the gospel" (1 Corinthians 9:16). But Paul also believed and acted upon God's promise that those who follow in obedience to His ways will gain rewards so great that they make any suffering on earth insignificant by comparison (2 Corinthians 4:17). Paul looked to the reward, saying he was running his race of life to win the prize that only God can give (1 Corinthians 9:24). 

The way for Paul to win the prize of God's approval is to make an offering of the Gentiles to God. His hope is that his work will be acceptable to God. Only God will evaluate the deeds of His people. If the works tested in fire remain, then each believer will receive a reward for what they have done (1 Corinthians 3:11-17; 2 Corinthians 5:10). It is inferred that Paul's ministry is not only to win the Gentiles to initial faith to be justified in God's sight, but also to disciple them to walk in faith. 

Paul desires that the Gentiles he disciples be sanctified by the Holy Spirit. In this context, this speaks of being set apart (sanctified) from the self-serving, exploitive ways of the world, and instead to walk in the love-your-neighbor-as-yourself ways of God. This fits perfectly with explaining why Paul would make the effort to write this letter to people he had not met. It was a part of his striving to fulfill the Great Commission Jesus gave His followers to make disciples of all nations (Matthew 28:19). 

As Paul states in 1 Corinthians, those who love God will gain rewards beyond their imagination (1 Corinthians 2:9). To love God is to keep His commands (John 14:15). To Paul, this "eternal weight of glory" of the future promise of reward is tangible because he can see it by faith (2 Corinthians 4:17; Hebrews 11:1). Paul is motivated to see the Romans follow God. He is not trying to get them to follow him.