What does Romans 15:3 mean?

Paul is speaking directly to those who are strong enough in their Christian faith that they feel secure in the grace of God. Such believers feel free to enjoy things formerly restricted for them by the law, including meat, wine, and not observing special religious days. In a broader sense, these are believers who don't feel led to add additional "do not" restrictions beyond those made clear in Scripture.

Paul is saying two important things about this attitude of Christian liberty. First, the "strong" believers are right to recognize that nothing is, itself, unclean (1 Timothy 4:4). Second, these believers should be willing to forego that freedom for the sake of those who are not yet strong enough in their faith to participate in those things (Romans 14:1–2).

Put more bluntly, Paul said in the previous verse that they should put pleasing their neighbors above pleasing themselves. After all, Paul now writes, they are following Christ. Christ did not please Himself in this life. He lived a life of self-sacrifice in serving and pleasing others.

Paul quotes from Psalm 69:9, applying it to Jesus. In that context, the reproaches—the mockery and criticism—of those who reproached God the Father fell on Christ. By comparison, Paul seems to be saying, strong-faith Christians should be willing to give up meat, or give up wine, or to skip the Sabbath, or any other matter of their personal freedom, for the sake of building up their weaker siblings in Christ.