But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— (ESV)
When Paul wrote these words it had been many centuries of the Jews having the Law. The Gentiles had the Law of God on their hearts. Neither the Jews nor the Gentiles had ever been successful in keeping the Law of God, not one person. The Law of God showed mankind what God demanded, but man could not fulfill it.
But now. Paul had just written in the previous verses how incapable man was of righteousness on his own. He even went so far as to declare that no human being will ever be made right with God by keeping the Law. That is really bad news. Paul had to tell the bad news before now telling us the good news.
The good news is that there is a new way to be made right with God. This new way of righteousness is apart from the law. This new way is Jesus Christ. He is fully God, yet he took on human flesh. He was the promised Messiah that the Law and the Prophets bear witness to. In fact, if the Jews would have understood it, the sacrificial system in the Law all pointed to Jesus Christ.
The Law of God brings judgment to every man and woman. Jesus steps into the situation and offers to keep the Law of God for us. In his life on earth he fulfilled God's Law perfectly. He then went to the cross to pay the penalty for everyone who sinned against the Law. Those who place their faith in him and receive his substitutionary work are eternally saved and have this new righteousness.
This "new" way of righteousness through Christ is the best possible news for mankind. Have you accepted Christ's work on your behalf? Or are you unsuccessfully trying to be "good enough" by keeping the Law?