This verse shows part of the Torah's purpose to reveal sin. (To the Jew first, to whom Torah was given, however, not only for their sake, but also for the sake of the gentile world, as the Jews were to be a light unto the world with this knowledge - i.e., Matthew 5:13-16).
The Torah makes man more "aware" that he is sinning (i.e., "that offence might abound"), as he is now violating "known and definable" commands from God. It is important to understand that God gave His Torah because of His mercy as an instrument to salvation (with faith/trust being the first of the commandments).
(One of the difficulties with Christianity is that it does not present a definition of sin. References are made to "disobeying God," or "going against the will of God," but what that means is very much left up to people to decide. Christianity's various denominations pick and choose what commandments from the "Old Testament" they believer are "applicable" (to their particular doctrines), while they all maintain the teaching that believers are "not under the law.")
The Torah, by revealing sin as such, is not "part of the problem," but rather, is "part of the cure." The Torah also "stirs up" sin in man, as by "laying down the law," many, who see themselves as above God, will respond indignantly, with deliberate violation.
Compare this to the parable in Mark 12:1-9, and how the tenants of the land respond to the owner's gestures. The more they are made aware of their improper behavior, the angrier they get:
Mark 12:1-9 - And he began to speak unto them by parables. A certain man planted a vineyard, and set an hedge about it, and digged a place for the winefat, and built a tower, and let it out to husbandmen, and went into a far country. And at the season he sent to the husbandmen a servant, that he might receive from the husbandmen of the fruit of the vineyard. And they caught him, and beat him, and sent him away empty. And again he sent unto them another servant; and at him they cast stones, and wounded him in the head, and sent him away shamefully handled. And again he sent another; and him they killed, and many others; beating some, and killing some. Having yet therefore one son, his wellbeloved, he sent him also last unto them, saying, They will reverence my son. But those husbandmen said among themselves, This is the heir; come, let us kill him, and the inheritance shall be our's. And they took him, and killed him, and cast him out of the vineyard. What shall therefore the lord of the vineyard do? he will come and destroy the husbandmen, and will give the vineyard
unto others.
Paul explains the "dual role" of the Torah in God’s purpose. To make known the full magnitude of sin (so that man could see it for what it was), and to act as an instrument of God’s mercy and grace
(the law is holy, just and good - Romans 7:12), in that it leads men to God.
This "duality" is in accordance to what God said when He gave the Torah:
Deuteronomy 11:26-28 - Behold, I set before you this day a blessing and a curse; A blessing, if ye obey the commandments of the LORD your God, which I command you this day: And a curse, if ye will not obey the commandments of the LORD your God, but turn aside out of the way which I command you this day, to go after other gods, which ye have not known.
See here -
Romans 5:1-5:21