Jesus said in Luke 9:24 "For whoever wants to save his life will lose it, but whoever loses his life for me will save it." Again, in Luke 17:33 "Whoever tries to keep his life will lose it, and whoever loses his life will preserve it." I have often argued that his can be simplified to: "giving is living and living is giving." St. Paul, in Romans 6, is making the same argument just in a different way. Paul's argument is that we need to "die to self". "For if we have been united with him in a death like his, we will certainly be united with him in a resurrection like his" (Romans 6:5). To be reconciled to God we must have faith and trust in Jesus Christ. This faith and trust is the surrender of our striving and our willful rebellion against God's purpose in our lives. In this surrender to Jesus Christ as the Lord of our lives, we encounter a kind of death -- death to our ego, to our sense of self as independent from our creator.
The argument is a central Biblical argument. We were created by God to be the "object of God's love". All that we have all that we are and all that we dream of being is a gift from God. Our purpose in life is to respond to that love and to live life in this love relationship with our creator. However, one consequence of sin (go back to Adam and Eve), is that humans are now in rebellion against their primary purpose. We believe we can live life without God, we believe we can live life independent of our purpose. We can be free to rule ourselves and be the owners and controllers of all that we survey. It is as Milton describes in Paradise Lost. The words he puts in Lucifer's mouth is a perfect description: "better to reign in hell than serve in heaven". Paul's argument is that only by dieing to ourselves and becoming "slaves to righteousness" can we ever hope to be free and to freely live into the purpose for which we are created. The way we "die to ourselves" is by surrendering to the Lordship of Jesus Christ and living our our primary purpose.