A person's choices and character do not exist in a vacuum. The way someone lives directly reflects on the parents and teachers who guided them, serving either as a profound source of pride or a painful cause for public shame.
The primary approach among commentators is that the call for a child or student to actively acquire wisdom [רש״י, אבן עזרא, עמנואל הרומי] serves two distinct purposes: to bring joy to the mentor's heart and to offer protection against insults. When a child acts foolishly, it brings disgrace upon the parents. They are often forced to remain silent when facing critics who accuse them of poor parenting, terrified that speaking up will only expose their child's flaws further. In contrast, a successful and wise child gives the parent a voice and the confidence to answer those same critics [רלב״ג, מצודת דוד, אבן עזרא, ביאור שטיינזלץ]. Even if a child was previously foolish and caused their parents to suffer the harsh words of those who shame them [ביאור שטיינזלץ], choosing a path of wisdom from this point forward shifts reality. It empowers the parents to push back against past accusations and prove the critics wrong [מלבי״ם, עמנואל הרומי].
On a specific historical level, this dynamic played out between King David and his son Solomon. David's enemies insulted him, claiming that his marriage to Bathsheba was rooted in sin. However, the fact that Solomon grew to possess such extraordinary wisdom served as living proof that their union was pure and desired by God from the very beginning of creation. Such supreme wisdom could never have emerged from a sinful act. Solomon's brilliant character ultimately provided David with the definitive answer to silence his detractors and prove their claims false [חומת אנך].
Beyond the human relationship, this dynamic also exists on a spiritual level as a plea from God to the sinner. Since a person only sins when a spirit of foolishness enters them, God asks the individual to become wise, abandon their foolishness, and repent. A person might hesitate, fearing that their troubled past means God will not truly welcome them back, and that accusing angels will argue that accepting their repentance is an act of unjust favoritism. To this, God promises that He finds immense joy in a person's return. It is precisely this sincere repentance that allows Him to answer the heavenly accusers and completely justify welcoming the sinner back with open arms [אלשיך].