In the context of the following verses, Bible dictionaries define reproach as shame, dishonor, injury, or even insult.
Therefore, let us go forth to Him, outside the camp, bearing His reproach. ….. And you, who once were alienated and enemies in your mind by wicked works, yet now He has reconciled in the body of His flesh through death, to present you holy, and blameless, and above reproach in His sight… (Hebrews 13:13 and Colossians 1:21-22 NKJV)
What do we make of this? Simple: When we bear the reproach of Christ, He will present us above reproach in the end.