Musa picked up the Tablets when His Anger subsided
Allah said next,
(And when calmed) and subsided,
(the anger of Musa) with his people,
(he took up the Tablets), which he had thrown out of jealousy for Allah and anger for His sake, because of his people worshipping the calf,
(and in their inscription was guidance and mercy for those who fear their Lord.) Several scholars of Tafsir said that when Musa threw the Tablets on the ground they were shattered and he collected the pieces afterwards. Musa found in its inscription guidance and mercy, but the specific details of the Law was lost, so they said. They also claimed that the shattered pieces of the Tablets still remained in the treasury safes of some Israelite kings until the Islamic State came into existence. Only Allah knows if these statements are true.