Paul exhorts us to consider a principle that applies to two areas of priestly conduct, offering sacrifices and praising God. In Greek, the emphasis in verse 1 is on the word "therefore." Why? Because Romans 12:1 begins the summary of the practical application of the principles, the teachings, the instruction that Paul gives in the first 11 chapters. He is preparing to draw practical conclusions to all the doctrinal things he taught, that is, about justification, sanctification, our calling, and the Holy Spirit, including the glorious things about Israel and what will happen in the future. It is as if he is saying, "In light of everything that I have given you thus far, here's what you are to do, priests! Sanctify yourselves by being a living sacrifice."
He means that, due to God's great gifts to us, it is "reasonable" for God to expect us to sacrifice ourselves. Sometimes modern Bibles will render this word as "spiritual," which is a correct alternative. The Greek word underlying it appears only one other time in the Bible, and in that place it clearly means "spiritual." Here, though, it means "reasonable," that it is right that we should be a sacrifice. It is logical, rational, to do this.
The "therefore" brings an exciting conclusion to his thought. He is saying that everything we do, every activity in all of life, is to be an act of worship in service to God. Everything! We are to live our lives as living sacrifices—except on those days that we want to do our own thing? No, not at all. God owns us, and He wants our lives lived all the time—every day, 24/7—as a living sacrifice. We are to sacrifice our lives to Him because it cost the life of His Son to give us the privilege of drawing close to Him.
So, He has every right to ask this of us. It is logical. It is rational for Him to demand it of us. In this case, even as Jesus is portrayed in the offerings of Leviticus 1-7, we, as His brethren, are both the priest making the offering and the offering that is being sacrificed.