David received the prophesy that he would not lack a son to sit on his throne. His son Solomon ruled in Jerusalem after him, then Rehoboam (under whom the kingdom was split), then Abiyam, all the way down to the exile. The House of David is the dynasty. The dynastic hopes of Israel hinged on physical lineages, so much so that Ezra recorded that there was a genealogical registry to validate the tribes of those returning to Israel under the Persian release (cf. Ezra 2:62 where the motivation was priestly).
The Gospels pick up where the stories of these kings leave off. There was the expectation of David’s son to rule again from Jerusalem (e.g., Ezek 37:24; Hosea 3:5), and this is the blueprint behind the genealogy of Matthew. The New Testament begins where I and II Kings leave off — it is III Kings: The Acts of the God-King who completes the Trilogy in flesh. This is helpful for interpreting the bible as one moves forwards and backwards in the scriptures. As the kingdom progresses forward, reading Samuel, Kings and Chronicles is an inherently Christ-centered enterprise. Then, picking up with Jesus and looking back, the genealogies of Jesus summarize (in name form) the whole history and whole story of the Old Testament. In this way, hearing and reading the Old Testament gives way to worship. When I read the accounts of the kingdom and her kings of old, my heart turns to King Jesus, of whom my faith lays hold.
The House of David is an unbroken chain of dynastic succession that passes cleanly through generations for hundreds of years (tracing from father to son), and climaxes in the great rule of God in human history; this rule is theological because it is Christological.
The genealogies of Matthew and Luke take up the whole sweep of all reality, and show how Christ is the climax of all history, all kingship, and all meaning. And now we have found the Son of David who is enthroned forever.