Out of Egypt: God's rescue of the Israelites from slavery in Egypt is a picture of his rescue of the follower of Christ from sin and death
The Old Testament (written four hundred years or more before Jesus was born) points forward to Christ. It does this through individual stories, like the story of Abraham and Isaac in Genesis chapter 22. But the writers of the New Testament also see the larger narrative of Israel's deliverance from slavery in Egypt as a picture of God delivering his people from sin and death and judgment through Jesus. The sacrifice of the Passover lamb is also taken up in the New Testament as a picture of the death of Christ.
The story of the deliverance from Egypt is found in the Old Testament books of Exodus, Leviticus, Numbers and Deuteronomy. The 'founding fathers' of the nation - Abraham, Isaac, and Jacob, had long passed from the scene, and their descendants had been immigrants in Egypt for hundreds of years. Gradually, as they grew more and more numerous, the Egyptian kings had progressively enslaved them, until, after about four hundred years, God raised up the great deliverer Moses, who led them out of Egypt, and gave them a 'covenant' relationship (agreement) with God, which was expressed in the Old Testament Law, especially in the Ten Commandments.
Pharaoh, the ruler of Egypt, was completely unwilling to let the Israelites go, and the Bible records a series of dramatic and miraculous plagues which God visited on the Egyptians, culminating in the deaths of every first-born son. Only then would Pharaoh release the Israelites.
The deliverance from Egypt, and the making of the covenant, were the most important events in the whole of Old Testament Jewish history. The Israelites looked back on these events as the founding of their nation
At the same time, God gave the Israelites a symbolic feast - the Passover. In the Passover, a flawless lamb had to be killed (i.e. sacrificed) and its blood sprinkled on the doorposts of the house. When the 'angel of death' saw the blood on the door frame, he would 'pass over' that house, and not kill the firstborn son. Thus by keeping the first Passover, they escaped from the judgment that fell on the Egyptians.
The New Testament sees the Israelites' deliverance from Egypt as being a model or picture of the much greater deliverance that God provides for his people so that they can escape from sin and death and judgment.
The New Testament specifically picks up the Passover lamb as a picture of the sacrificial death of Jesus for the sins of his people - see, for example, 1 Corinthians chapter 5 verse 7. It is not a coincidence that according to the Gospels, Jesus' death took place at the time of the Passover. Next.
For more about how the whole Bible points to Jesus - Old Testament as well as New Testament - read Vaughan Roberts' excellent 'God's Big Picture: tracing the story-line of the Bible', or Graeme Goldsworthy's 'According to Plan: the unfolding revelation of God in the Bible', or Alec Motyer's 'Look to the Rock: An Old Testament Background to our understanding of Christ.'