Before the law of Moses there was sin in the world but there were not exactly sins. Sin, in singular, is that condition of enmity and separation from God that affects all mankind, due to Adam as father and representative of all humanity, because he didn't obey God's commandment. But sins, in plural, are the individual transgressions of God's law. Sins are personal, and they affect each individual in particular. Sin is universal and affects all mankind, except Jesus Christ.
Sin is something that has passed to all human beings, because all sinned, as the quotation says.
The only exception to this universal law mentioned in the Bible, is Jesus Christ. Before the law of Moses there was sin in the world but there were not exactly sins. Sins, on the other hand, as we have said, are personal transgressions of the law of God. But God didn't promulgate any law, that we know, until Moses. The consequence of this is that human beings who lived between Adam and Moses didn't have a law to obey or that they could break.