Moreover the law entered, that the offence might abound. But where sin abounded, grace did much more abound:
Note 8 at Ro 5:20: Paul was writing to Jewish Christians who had mistakenly thought that faith in Christ alone was not enough to produce justification. They thought one also had to fulfill a minimum standard of holiness by complying with certain commands of the Old Testament Law. That's what occasioned Paul's whole teaching on justification by faith.
Paul had so conclusively proven justification by faith in Christ alone that he knew the legalistic Jews were wondering, "So, what was the purpose of the Law?" He stated that purpose in this verse. The Law was given to make sin increase, or super-abound ("abound" - "PLEONAZO" - "to do, make or be more, i.e. increase...to superabound" [Strong's Concordance]).
As explained in note 4 at Ro 3:19, the purpose of the Law was not to strengthen us in our battle against sin but to strengthen sin in its battle against us. Sin had already beaten us; we didn't know it. The Law brought that realization to us so that we would quit trusting in ourselves and call out to God for salvation.
So the Law made sin and all its devastating effects abound, but God's grace abounded even more. The Law gave sin so much dominion against us that the grace of God is the only way out.