What does it mean to judge righteously

It conveys the idea that judgment is made solely on the basis of merit. In other words, you should judge someone based on their actions or behavior, rather than their race, gender, or social status.

Judging righteously would mean not showing undue favor or disfavor towards someone without a just cause. Jesus would be the righteous judge in the future. This implies that Jesus would bestow the gifts of God on those who are innocent and punish those who are wicked.