Romans 5:20-21 meaning

In verse 13, Paul told his audience that "the law" has made "transgression [sin] increase" because people were made more aware of things that God wanted them to do or not do. Here, Paul is summarizing this point again by showing the audience that when the law came it increased sin because it made people understand when they were sinning.

If a child was never told that he could not take cookies from the jar, then taking those cookies would not be wrong. However, if the child was told those cookies are not to be taken, then each time he took a cookie, it would be in disobedience.

However, Paul points out that although the law increased sin, through Jesus's sacrifice grace abounded all the more to cover those sins (v 20). Certainly sin increased when the law was given, but God's grace abounded to a sufficient degree to cover all transgressions; every sin that has been or will be committed was nailed to the cross with Jesus (Colossians 2:14). The Law came in so that the transgression would increase; but where sin increased, grace abounded all the more (v 20).  

Sin reigned in death after Adam's sin. But Jesus's sacrifice restored righteousness, received through faith, because His sacrificial death abolished death (2 Timothy 1:10). This sacrifice from the God-man Jesus settled the payment for our sins, and through faith in Jesus Christ we can have eternal life instead of the death (separation) that sin brings: So that, as sin reigned in death, even so grace would reign through righteousness to eternal life through Jesus Christ our Lord (v 21).

Grace will reign in our lives when we choose to walk by faith that God's ways are for our best. This leads us to live righteously, walking in God's (good) design. This means choosing to love others and serve their best interest rather than following the exploitive ways of the world.

We gain the free gift of eternal life through Jesus Christ our Lord through faith, not by following rules (John 3:14-15; Romans 4:1-3). We also gain the experience of eternal life as well as the reward of eternal life in the same way, through Jesus Christ our Lord. It is through His power that we can regain the life that was lost in the Garden of Eden.

God's grace frees us from sin and empowers us to live obediently toward God. It is through living in the power of Jesus that grace would reign through righteousness. To live in this restored harmony with God and His ways leads to a life of unparalleled quality. Through Jesus, we gain the newness of life, the experience of eternal life made possible by Jesus Christ our Lord, who reconciled us to God, giving us access to God's abounding grace over our lives.