Jesus has just finished telling the disciples seven parables about the nature of the kingdom of heaven and asks if they understood them. I have no reason to be skeptical of the disciples' answer, "Yes." Some commentators think this was arrogance and presumption talking, but it seems to me that the disciples have been honest with Jesus about their ignorance (see verse 36 of this chapter), and after Jesus gives them further explanation he accepts their claim that they have finally understood him. Sure, maybe they haven't yet grasped the full implications of this kingdom and how much they will have to suffer for it, but they certainly have understood more than the multitudes.
Compare your own understanding of these parables to that of a non-believer. Does a non-believer seek to understand these parables? Does he view them as profound? Does he meditate on their meaning? Does he surf onto a blog to read someone else's explanation of them? See, that's the difference. And while you might not claim to understand the full implications of these kingdom parables, you do see how these stories of seeds and harvests and treasures and trees and dough and fish relate to the spiritual realties Jesus is talking about, right? If you see the connection then it's safe to say that you, too, have understood. You are a disciple.
When Jesus says, "Every scribe who has become a disciple of the kingdom," he is talking about someone who has made the transition from the old covenant to the new. A scribe is not merely a clerk but a teacher of the old covenant law. And so a scribe who becomes a disciple of the kingdom is someone who possesses true insight into the law, who understands that these things are mere shadows pointing to the substance of Christ. This scribe becomes a disciple of the kingdom because he sees that Jesus is the fulfillment of all the old covenant promises, and the answer to every longing and lament that accompanied Israel's failures.
I haven't figured out why a head of a household would be someone who brings out of his treasure the old and the new. Is he a patriarch whose life spans several generations, which is why he finds himself in possession of both old and new things? No one has bothered to explain this to me, so that is my best guess. Both old and new treasure have their special value, just like the old and new covenants. The one who treasures both can use his old-covenant-scribe knowledge to shed light on his new-covenant-disciple understanding. He can show how promise led to fulfillment, how type pointed to reality, how hope sprung from failure and grace was born out of betrayal.