In the previous verse, Jesus is mentioned as being willing to suffer for the sake of future joy. Paul begins a new section in Romans 12 of his letter. Here, he begins describing how we should live as believers in Christ. Paul emphasizes the need for Christians to offer themselves as living sacrifices to God. To put it another way, the only rational response to God's mercy in giving us eternal life is to offer up our lives as a sacrifice for His purposes.
As fallen sinners, we are beyond comparison in terms of the grace and mercy we have received from God. It is entirely due to God's wonderful grace that we are saved. As humans, there is nothing we can do to repay God. However, we can certainly give ourselves to Him in gratitude and love by presenting our bodies as living sacrifices that are pure and pleasing to God.
Obviously, this is not a command since God desires that our love and service be freely offered. Nevertheless, in light of the inestimable riches of God's free grace, which He has poured out in great measure on fallen sinners, surely presenting our bodies, day by day, in spiritual worship is an honor and a delight for ourselves.

Dear Heavenly Father, You have given me eternal life through Your Son Jesus Christ as a gift of forgiveness of my sins. Now I offer You everything I am and all I have. That is my logical response to the love You have shown me. This is my way of worshiping You spiritually. My life is consecrated to You. Please take my life, my Lord, and allow me to be Your living sacrifice. It is my joy to praise You for the amazing gift of salvation. 
In Jesus's name I pray. Amen.