Because Jonathan's dead, and David still feels bad about it—considering that they were two members of the "Super Best Friends 4eva Club"—he wants to do something nice for any remaining members of Saul's family.
The servant of Jonathan's son Mephibosheth (the baby who had been dropped by his nurse and left lame earlier in the story), Ziba, says that Mephibosheth is just the right person to help.
David has Mephibosheth brought before him, and Mephi' bows and does obeisance to David. But David tells him not to be afraid: he's going to take care of him, and let him live in his house and eat there. He'll also give the lands originally belonging to Saul to him, as well.
Mephibosheth is totally floored by this, and says he can't believe he deserves such kind treatment.
David tells Ziba that he (Ziba) and his sons will till the land for Mephibosheth and provide him with food, while Mephibosheth himself will dine at David's table. Also, the text randomly mentions that Mephibosheth himself has a son named Mica. Thanks for the info, text.
So, all this happens: Ziba and his household till Mephiboseth's land and provide him with vittles, while Mephiboseth lives it up in high style with David.