I want to expand a bit on the other good answers already given.
One of the first things to note is the state of the people of Israel at the time. They were enslaved to Egypt.
So they ruthlessly made the people of Israel work as slaves 14 and
made their lives bitter with hard service, in mortar and brick, and in
all kinds of work in the field. In all their work they ruthlessly made
them work as slaves. Exodus 1:13-14 ESV
Jesus taught that we, as sinners, are also slaves--not of Egypt, but of sin.
Jesus answered them, “Truly, truly, I say to you, everyone who practices sin is a slave to sin. John 8:34 ESV
The Passover Lamb
God instructed the Israelites to select an unblemished lamb, keep it for four days, then slaughter it at twilight.
Tell all the congregation of Israel that on the tenth day of this
month every man shall take a lamb according to their fathers' houses,
a lamb for a household... 5 Your lamb shall be without blemish, a male
a year old. You may take it from the sheep or from the goats, 6 and
you shall keep it until the fourteenth day of this month, when the
whole assembly of the congregation of Israel shall kill their lambs at
twilight. Exodus 12:3, 5-6 ESV
John the Baptist introduced Jesus as the Lamb of God. Though the timing is disputed, it does seem that Jesus entered Jerusalem on Palm Sunday, stayed for four days, and then was crucified, dying close to the end of the day. Pilate himself pronounced Jesus as unblemished as well.
The next day he saw Jesus coming toward him, and said, “Behold, the
Lamb of God, who takes away the sin of the world! John 1:29 ESV
Then Pilate said to the chief priests and the crowds, “I find no guilt in this man.”
Luke 23:34 ESV
Salvation and Impending Judgment
God told the Israelites to observe the Passover in order to escape the impending judgment that He was going to execute upon the land.
For the Lord will pass through to strike the Egyptians, and when he
sees the blood on the lintel and on the two doorposts, the Lord will
pass over the door and will not allow the destroyer to enter your
houses to strike you. Exodus 12:23 ESV
Plain and simply, those who heard, believed, and acted on that belief by observing the Passover and applying the blood were spared from God's judgment. Those who did observe the Passover received the judgment of God.
The New Testament teaches that there is an impending judgment for all mankind as well. Jesus taught that there is only one way in which to escape this judgment, and that is by faith in Jesus, our Passover Lamb. Those who hear, believe, and place their trust in this means of escape will indeed be saved from judgment. Those who do not respond to this will, indeed, receive God's judgment.
"because he has fixed a day on which he will judge the world in righteousness by a man whom he has appointed; and of this he has given assurance to all by raising him from the dead.” Acts 17:31 ESV
“For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. 17 For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. 18 Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. John 3:16-18 ESV
The Israelites were commanded to eat all of the lamb. Jesus may have been alluding to this in John 6 when He spoke of eating His flesh.
The Israelites were specifically commanded to eat the Passover meal "in haste". We eat quickly when we are in a hurry to leave, and they did the same. They had been enslaved for many years, but by eating in a hurry, they exhibited faith that they would soon be set free.
The people were commanded to apply the blood to the doors of the homes. Without this application of the blood, there would be no escape from judgment. The blood of Jesus is the means of our escape. It's interesting that it wasn't the righteousness of the people inside the houses that made them worthy of escape, but the blood of the lamb that was on the door of their houses.
In this manner you shall eat it: with your belt fastened, your sandals on your feet, and your staff in your hand. And you shall eat it in haste. It is the Lord's Passover. Exodus 12:11 ESV
The Last Supper
Jesus was crucified during the Passover Feast. The Last Supper may have even been a Passover meal. If so, the cup He lifted up would possibly have been the one called the "Cup of Redemption". He may have died while many Jews were sacrificing their own chosen lambs.
It is interesting that during modern day Jewish celebrations of the Passover, there are three pieces of bread placed together in a white cloth. The middle or second one is removed during a part of the ceremony, broken, then hidden away in another white cloth until it is brought out later.
Jesus, the Second Person of the Trinity, spoke of His body being broken. If the meal was, indeed, a Passover celebration, this may have actually been the time that He took and broke the second piece of bread. Jesus's body was broken, and He was hidden away in the tomb for three days, but resurrected later.
Heaven, the Promised Land
The exodus that began at the Passover Feast culminated in the entrance into the Promised Land--a distant land "flowing with milk and honey". God had indeed promised them that He would bring them there, and He did. The exodus of Christians begins when we put our faith in the blood of Jesus, and it culminates when we reach a distant land called Heaven.
Let not your hearts be troubled. Believe in God; believe also in
me. 2 In my Father's house are many rooms. If it were not so, would I
have told you that I go to prepare a place for you? 3 And if I go
and prepare a place for you, I will come again and will take you to
myself, that where I am you may be also. 4 And you know the way to
where I am going.” 5 Thomas said to him, “Lord, we do not know
where you are going. How can we know the way?” 6 Jesus said to him, “I
am the way, and the truth, and the life. No one comes to the Father
except through me. John 14:1-6 ESV