And now I know well that thou shalt surely be king These wonderful dispensations of Divine Providence over David, and David's no less wonderful virtue, at last convinced Saul that God designed him to be the king of his people, and that none could hinder his establishment. That thou wilt not destroy my name, &c. As it was usual for kings to do in those days, generally destroying the families of those to whose thrones they were advanced.
Related Commentaries of 1 Samuel 24:20
1 Samuel 24:20-21
20 And now, behold, I know well that thou shalt surely be king, and that the kingdom of Israel shall be established in thine hand.
21 Swear now therefore unto me by the LORD, that thou wilt not cut off my seed after me, and that thou wilt not destroy my name out of my father's house.