Paul is explaining a couple things in this passage. He is telling us how sin entered the world. He is contrasting Adam and Jesus. He points out, to his Jewish audience , that the Law points out our sin; rather than a cure it’s a tool, like an XRay to show us our inner nature. (and the news is not good).
Paul starts out by saying through Adam we have received a sin nature. Because of this, death came to all of us. Recently I read a medical thriller in which one man had engineered a very contagious killer virus and the race was on to prevent the spread of the virus. Because of Adam, Paul is saying, we are infected.
Paul then explains that the Law, when it came, is our diagnostic tool to show us the extent of our problem. He contrasts the deed of Adam and the work of Christ; Adam sinned once and many were condemned, whereas the gift of God (the work of Christ) followed many sins and brought grace to all. Paul says God’s gift is abundant, a cure freely given due to God’s love.
Another contrast which Paul makes is death and life; he begins the passage by saying that through Adam, death came to all men. In the last sentence of the chapter, (verse 21) he wraps up by saying that we have eternal life through Jesus Christ our Lord. This then, is our hope and comfort.