The RSV and ESV change the sense of the second half of the proverb by rendering the word ‘cover’ as ‘ignore’, and the word ‘shame’ as ‘insult’. The meaning is then that a prudent man does not react like the fool whose anger erupts immediately, but instead acts as if no insult had been given and does not respond; he ignores the shame coming from outside him.
Now, you see, this is all in the context of words. Words are not actually mentioned here but, bearing in mind it is in the block of proverbs that is about words, it helps us to rightly interpret it. If we did not bear the context in mind in our interpretation we would say, oh, this is about whether you react hastily or not. This is about whenever you are troubled or insulted, don't let fly but just control yourself. But the proverb is actually saying more than that. When you are in a discussion, or you are interacting with another person, witnessing may be, or trying to discuss a point of truth and help another, if the waters are muddied by contradiction or something like that, the last thing to do is to regard as the important matter my honour. It relates to the previous proverb.
Now the wise man will say, Look, the purpose of this conversation and interaction or witness is truth. It is not to defend my honour. If the person I am talking with unworthily or unreasonably puts me down on some small point, or some big point, I am not going to divert or chase my honour, justify myself, defend myself, react, retaliate, take verbal vengeance. That is not the issue, the issue is we were trying to clarify the truth, so I am going to keep to the point, keep cool. The fool loses sight of what the discussion was about to make a point for the truth and he is side-tracked straight away into defending his honour and reacting, and perhaps verbal vengeance. Don't let your feelings get too involved. Stick to the issue. We are bringing about the honour of the Lord, the salvation of precious souls by his mighty grace, the helping and deepening of one another, the bringing up of our children and the goal is important – the truth and the words.