David shows King Saul that he cuts the hem of his cloak.
After chasing down the Philistines, King Saul returns to hunt down David. While in the wilderness, Saul decides to go inside a cave. Unbeknownst to him, David and his men were actually hiding out in this very same cave.
So, David basically has Saul by the balls here. He can easily take him out if he wanted to. However, David something interesting. Instead of killing the king, he takes his sword and secretly cuts a piece of his robe off. King Saul is unaware that this happens, and after some time walks out of the cave.
In the daylight, David follows and calls out to Saul, pointing out that he could have taken his life but instead cut his cloak. Saul is touched by this show of mercy, and admits that David is the one who has been anointed by God and that he will be king.
Facing this inevitability, Saul asks David to have mercy on his bloodline and to let his descendants live, which David agrees to.
Again, we are faced with the strange mentality of these people. Rather than put them down for it, it's interesting to see it from their eyes, as there are still people out there today who have similar ideas about how the world is organized.
David's show of mercy is great, and Saul's acceptance of David is even better. But just the thought that one's descendants are at risk from an up and coming king is a strange artifact of the mentality of monarchies. Eventually, Saul's descendants may very well have a claim to the throne.
David is young and has not been groomed for royalty, so he's probably naive to this point. So, David doesn't care about that and agrees to Saul's terms. After all, David is Yahweh's "anointed one", right? What's the worse that could happen?