It is the Law, from which Paul has just quoted, that condemns the Jews so that all men will be silenced on the Last Day and the whole world held accountable to God.
We look at the attitude of the Jews, who thought that mere possession of the Law would save them. In contrast, Paul shows that the Law could never save - not because there was anything wrong with the Law but because of the sin of the people.
Unbelievers are urged not to seek their salvation through works but only in Christ. Believers are exhorted to keep the Law, not to earn righteousness for salvation but because it pleases and honors God and those who love Him will keep His commandments.