Paul begins by saying that, since we have been justified by faith, we have peace with God through our Lord Jesus Christ. Through him we have also gained access to the grace in which we stand, and we rejoice in hope of the glory of God. More than that, we rejoice in our sufferings, because suffering produces endurance, endurance produces character and character produces hope. That hope does not put us to shame, because God's love has been poured into our hearts through the Holy Spirit.
In verses 6 to 11 Paul shows the proof of that love: Christ died for us at the right time, when we were weak, ungodly, sinners and even enemies. One would hardly die for a righteous person; God shows his love for us in that Christ died for us while we were still sinners. If we were reconciled to God by the death of his Son, much more shall we be saved by his life, and so we rejoice in God.
From verse 12, Paul compares two men. Through one man, Adam, sin entered the world and death through sin, and death spread to all. Death reigned from Adam to Moses, and Adam is a type of the one who was to come. But the free gift is not like the trespass: if many died through one man's trespass, much more has the grace of God abounded for many through the one man Jesus Christ. One act of righteousness leads to justification and life for all who believe, just as one man's disobedience made many sinners and one man's obedience makes many righteous.
The law came in to increase the trespass, but where sin increased, grace abounded all the more, so that, as sin reigned in death, grace might reign through righteousness leading to eternal life through Jesus Christ.