The New Testament authors understood Jesus to be the culmination of the Old Testament. He is the Last Adam, true Israel, the suffering servant, the son of David, the faithful remnant, the ultimate prophet, the reigning king, the final priest.
Here is a good, concise summary of the Israel/remnant theme from a New Testament perspective:
. . . Jesus had become a remnant of one. He was the embodiment of faithful Israel, the truly righteous and suffering servant.
Unlike the remnant of the restoration period, he committed no sin (Isa. 53:9; 1 Pet. 2:22).
As the embodiment of the faithful remnant, he would undergo divine judgment for sin (on the cross), endure an exile (three days forsaken by God in the grave), and experience a restoration (resurrection) to life as the foundation of a new Israel, inheriting the promises of God afresh.
As the remnant restored to life, he becomes the focus of the hopes for the continued existence of the people of God in a new kingdom, a new Israel of Jew and Gentile alike.
As the nucleus of a renewed Israel, Christ summons the “little flock” that will receive the kingdom (Dan. 7:22, 27; Luke 12:32) and appoints judges for the twelve tribes of Israel in the new age (Matt. 19:28; Luke 22:30).
The church is viewed as the Israel of that new age (Gal. 6:16), the twelve tribes (James 1:1), “a chosen people, a royal priesthood, a holy nation, God’s special possession” (Ex. 19:6; 1 Pet. 2:9).
A sinful nation, Israel could not suffer vicariously to atone for the sins of the world. The sinfulness of the nation made it unacceptable for this role, just as flaws would disqualify any other offering. Only a truly righteous servant could bear this awful load.
—Tremper Longman III and Raymond B. Dillard, “Isaiah,” An Introduction to the Old Testament, 2d ed. (Grand Rapids, MI: Zondervan, 2006), 315.
The two best books I’ve read on this fulfillment theme are Hans LaRondelle’s The Israel of God in Prophecy: Principles of Prophetic Interpretation and David Holwerda’s Jesus and Israel: One Covenant or Two? (Keith Mathison has a good review of Holwerda’s volume here.)
Jesus is the true Israel, and the church becomes the Israel of God as it unites to True Israel. The same is true for ethnic Israel, whom God has not abandoned. But their only hope is to be united with Jesus, the ultimate suffering servant.