What does Romans 15:16 mean?

Paul is describing to the Christians in Rome the authority by which he writes. This is the source of his knowledge about the grace of God and how to live the Christian life on earth.

God is the one who has appointed Paul to be a minister of Christ to the Gentiles. He is focused on one message, the gospel—the "good news." He acts as a priest might in service of the gospel of God, representing the truth of God's grace to them on God's behalf. Paul is not a priest, however, but an apostle. In fact, because of Paul's Christ-given role as an apostle, and that he wrote under the inspiration of the Holy Spirit, his teaching in Romans stands as the very Word of God.

Paul adds his purpose in writing to them: so that the offering of their lives to God, as Gentile Christians, might be acceptable to God and sanctified by the Holy Spirit. Again, Paul is picturing the role of a priest offering a sacrifice to God. In that picture, he is the priest and their lives are the offering. He is working hard to make sure that offering, their lives, are acceptable to God.

Paul understands, of course, that though his teaching might be used by God to help shape the living-sacrifice lives of the Roman Christians into something acceptable to God, he is not the one who accomplishes that. The Holy Spirit is the one who supernaturally sanctifies believers, transforming us from the inside out (2 Corinthians 3:18).