The commentaries discuss why it was necessary for Moshe to “grab hold” of the Luchos before he broke them, considering that he was already holding them as he descended the mountain.
One explanation is that Moshe did so for legal reasons—to establish his exclusive ownership of the Luchos before acting on his decision to break them. Although the Torah states that G‑d “gave” Moshe the Luchos, and the Talmud interprets this verse to mean that the Torah was given to Moshe as a gift, Moshe had not intended to keep the Luchos for himself, but to confer their ownership on the entire Jewish nation. Moshe therefore “grabbed hold” of the Luchos before breaking them, to reestablish outright ownership over them and avoid the possibility of damaging (or stealing) property that did not belong to him exclusively.
Alternatively, Moshe’s repossession of the Luchos before breaking them stemmed from his extraordinary devotion to the Jewish people. By claiming exclusive ownership, Moshe sought to ensure that the guilt of shattering the holy Luchos would lie squarely on his shoulders, and not on the Jewish people, whose behavior caused him to do so.
—Likkutei Sichos, vol. 34, pp. 51–56