What does “not the putting away of the filth of the flesh” mean?

Answer:

Sometimes it helps to look at other translations. "Corresponding to that, baptism now saves you -- not the removal of dirt from the flesh, but an appeal to God for a good conscience -- through the resurrection of Jesus Christ" (I Peter 3:21 NASB).

Baptism is not about getting the outside of the body clean but about cleaning up the dirt of sin inside the heart. The Jews had a problem looking only at the physical and missing the spiritual implications. "Woe to you, scribes and Pharisees, hypocrites! For you clean the outside of the cup and of the dish, but inside they are full of robbery and self-indulgence. You blind Pharisee, first clean the inside of the cup and of the dish, so that the outside of it may become clean also" (Matthew 23:25-26). Christians should not fall into the same trap. In obeying God's command to be baptized you are appealing to God to give you a clean conscience. This happens, not because of some property of the water, but by yielding to God's command you come into contact, so to speak, with the resurrected life of Christ.

"Or do you not know that all of us who have been baptized into Christ Jesus have been baptized into His death? Therefore we have been buried with Him through baptism into death, so that as Christ was raised from the dead through the glory of the Father, so we too might walk in newness of life. For if we have become united with Him in the likeness of His death, certainly we shall also be in the likeness of His resurrection, knowing this, that our old self was crucified with Him, in order that our body of sin might be done away with, so that we would no longer be slaves to sin; for he who has died is freed from sin. Now if we have died with Christ, we believe that we shall also live with Him" (Romans 6:3-8).