We have the custom of reciting those psukim after ברכת התורה.
Rabbi Bashevkin asks, why those psukim? I understand why I want to learn some Torah after saying the ברכה, but why specifically those?
He starts with a completely unrelated Gemara:
Most of those answers I understand: Rabbi’s children are at risk for being more arrogant about their Torah learning, which is not conducive to becoming a תלמיד חכמים. But רבינא answers: they do not say ברכת התורה. Who doesn’t say it? The parents or the children? And what does that have to do with not becoming a תלמיד חכמים?
There is another gemara about ברכת התורה. Yirmiyahu discusses the reasons for the destruction of the בית המקדש:
The usual explanation for that is that the ברכה on תלמוד תורה is not simply acknowledging the מצווה, but appreciating the sweetness of learning:
The key is והערב נא, “may it be sweet”. We have to appreciate Torah לשמה, as a joy, not as a burden. If we don’t, that is considered עזבם את תורתי. And we point out in the ברכה that the only way Torah can be transmitted וצאצאינו וצאצאי עמך בית ישראל is if we experience והערב את דברי תורתך. Read the ברכה as "Sweeten the words of Torah in our mouths, and then we and our children will [a consequence, not a prayer] learn Torah לשמה.
And that is what can be lost among the children of תלמידי חכמים. They may recite ברכת התורה appropriately, but they don’t teach their kids to recite it, to see the Torah as sweet. They don’t teach their kids, not because they don’t know it is important, but because they have communal responsibilities and their own responsibility to learn Torah, and assume the children will learn by osmosis, just by being in the household of a תלמיד חכמים:
The prototypical example of this was Moshe’s own children:
So we have to love Torah. But there is a danger that we will lose sight of other things while obsessed with the Torah. The Maharal suggests that the love of Torah learning could eclipse the love of עבודת ה׳:
Rabbi Bashevkin suggests that just like אהבת תלמוד תורה may lead to a loss of אהבת ה׳, it may also lead to a loss of אהבת רעך. We lose the connection to other human beings. That was the problem with the תלמידי חכמים. The cure for that is ברכת כהנים.
So, he suggests, we recite the psukim of ברכת כהנים to remind us that אהבת תלמוד תורה is not the only goal.