Now we know that whatever the law says, it says to those who are under the law, that every mouth may be stopped, and all the world may become guilty before God. Therefore by the deeds of the law no flesh will be justified in His sight, for by the law is the knowledge of sin. (Romans 3:19-20)
After demonstrating the sinfulness of the Jews using the Scriptures, Paul emphasizes that the Jews could not use the Law to stand as righteous before God.
No one can say anything.
What Paul has shown the Jews from the Scriptures about their own sinfulness was designed to stop their talking about being superior in some way to the Gentiles.
The Gentiles had sinned and were guilty before God (Romans 1:18-32).
The Jews had also sinned and were guilty before God (Romans 2:1-3:19).
The very design of the Scriptures, in pronouncing certain things to be sins, and in declaring the Jews to be guilty thereof, is to silence them on the question of justification by law, to destroy every vestige of hope therein, and thus to shut them up to the justification which is in Christ. - Lard, Romans, 111
The Law of Moses, which they thought showed they would be treated differently, actually pointed out their sin and highlighted their guilt. The Jews were relying on the law for something the law could not do.
By following the Law and its commands, no person would be justified by God (see Galatians 2:16), and this applied equally to Jews and Gentiles. - Pollard, Truth for Today Commentary, 117
The Law does not justify.
Law simply points out what is sinful, and it gives consequences to those who break the law. All were guilty of breaking the law - whether it was the Jews breaking the Law of Moses or the Gentiles breaking the truth they had from God (Romans 1:18-21) - so all were condemned by the law.
We understand how this works. So long as we follow the laws of our country, we are innocent of any crime. If we break the law, we are guilty and deserve whatever punishment is handed down for breaking the law.
When we break the law, all the law can do is pronounce us guilty and pass sentence.
The simple fact is, justification cannot come through law, for law simply points out sin, and passes judgment when men are guilty of violating the law. The deduction is, therefore, that all men, being guilty of sin, were condemned by the law they were under. Consequently, there was the need for something that would provide redemption, something other than law. - Wacaster, Studies in Romans, 117-118
Being “justified” means to regard as just or innocent. You have committed a crime, yet you’re allowed to go free as someone who committed no crime.
However, Paul points out that someone cannot be justified by the law - because the law condemns the guilty, and all are guilty.
Something - or Someone - else is needed to “justify” lawbreakers.