To present your body as a 'living sacrifice' means to offer your entire self—heart, mind, and actions—to God as a daily act of consecrated worship. The 'renewing of your mind' is the ongoing work of the Holy Spirit, who shifts your perspective from the world's values to the will of God, enabling you to live for Christ.
In Romans 12:1, Paul uses the language of the Old Testament sacrificial system but flips it on its head. Under the old covenant, sacrifices were brought dead to the altar; here, you are to present your bodies as living sacrifices. This means your life—every action of your hands, the words of your mouth, and the movements of your feet—is to be set apart for God’s service, not as a means to earn favor, but as your reasonable response to the mercy you have already received in Christ. As the 1689 LBCF Ch.13 §1 teaches, sanctification is the work of the Spirit whereby we are renewed after the image of God, enabling us to die to sin and live to righteousness. You are not sacrificing yourself to become accepted; you are living a sacrificed life because you are already accepted and loved in Christ.
Transformation is not a matter of willpower or moralistic self-improvement. It begins with the 'renewing of your mind,' which Paul connects to being 'renewed in the spirit of your mind' in Ephesians 4:23. When you are 'not fashioned according to this world' (Romans 12:2), it is because the gospel has reoriented your desires. You are more flawed than you ever dared believe, but because of the finished work of Christ, you are now a new creature. The Holy Spirit takes the truth of the gospel and makes it the lens through which you see reality. You stop conforming to the age, not by gritting your teeth, but by being transformed from the inside out, finally proving what is the good, pleasing, and perfect will of God.