How does the Passover foreshadow the coming of Jesus as the Messiah?

It's a great question and I want to expand on some of the good answers given. Much of the powerful symbolism and foreshadowing is missed or not-fully understood by most Catholics and Protestant because they don't understand Jewish culture and the feasts. All the 7 feasts point to the Messiah, and no other feast foreshadows his death as perfectly as the Passover.
The Passover has been celebrated for thousands of years - even before Christ. Some of the long-held traditions were adopted even before Christ, but were not specifically commanded in the original event, when the Death Angel passed over the homes of those with the blood. First let's deal with the ways Christ death was foreshadowed in scripture and how it was fulfilled.

• First, the passover lamb, had to be a male - young, and perfect - without any spot or defect. They couldn't give an old sheep. Christ was perfect, and totally without sin, and his blood was shed for us.
• Second, Passover is linked with the feast of unleavened Bread, which is 7 days, and in during these 7 days - they could not use or even have any yeast/leaven in their homes. Yeast is a picture of sin. We can't hide sin in any area of our lives. There was a practical purpose for this, as cleaning out the yeast, prevented disease from mice and rats.
• All those who had sprinkled the lamb's blood on their doorposts were spared from the death of the firstborn. Even gentiles/Egyptians who put the blood on the door posts were also spared. [Christ's sacrifice is for the whole world- Jew and Gentile alike.]
• The Children of israel were told to use a branch of hyssop, - a plant used by the Priests. We also find in the crucifixion, that the soldier put vinegar on a sponge on a branch of hyssop to give to Christ on the cross.
• Christ was literally on the cross- crucified at the exact time that the Passover lambs were being slaughtered in the temple. Afternoon. Hmmmm. Over thousands of years, many special rituals have come into Passover and are kept by Jews around the world, and they point to Christ.
• The Matzo is pierced and broken. Look at a Matzo cracker. It is pierced with many holes. Then the Matzo is broken into three pieces. Hmmm. [Three members of the Trinity- Father, Son and Holy Spirit] Then, they pick the middle piece [Not the 1st/ top, or not the 3rd/bottom, - the 2nd person of the trinity is Christ.
• Then, this matzo is literally wrapped in cloth and hidden. Christ was wrapped in linen cloth and hidden away in a borrowed tomb.
• Then, a ransom is paid for this matzo. It is hidden, and children look for the special matzo, and a small coin or prize is given. This piece is called the Affikoman - and it literally means that which we look forward to. Christ is the one we look to, and look forward to his return.
• There are four cups of wine, and one of these is the cup of redemption. With his sacrificial death on the cross- Christ paid the ransom, redeemed us, and atoned for the sins of the whole world, reconciled us to God, and the last word on the cross was "Tetelestai". The translation in many English Bibles says "It is finished" but this is a very poor translation. There have been many archaeological discoveries from the 1st Century, and we know that this term was a widely-used term from accounting, and it literally means "Paid in Full".
  Christ was crucified on Passover, in the grave for 3 days and 3 nights, on Feast of Unleavened bread - as the perfect lamb without sin, and then rose on First Fruits.

I want to expand a bit on the other good answers already given.

Slavery

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

Other Notes

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

In short, if you recall the Passover was a celebration of the memorial of The Angel of Death 'passing over' the houses whose doorframes (wooden frames) were stained with the blood of the Passover Lamb. Inasmuch as Jesus' death does the same, that is, causes death to 'pass over' those that are marked with his blood, he is in effect 'our passover' both in the sense of being the Paschal Lamb whose blood averts death and that his crucifixion and resurrection are pre-figured by the Passover celebration itself.

To go a little deeper, the blood on the doorframe is symbolic of those 'participating in his death', and in evading the sting of death, they also 'participate in his resurrection' as the Hebrews' firstborn were also spared death. If we take the household vis a vis the Passover to be symbolic of the body of the person, the 'firstborn' of the family is the soul. Thus, those whose bodies are marked by Christ's death through baptism escape death, provided they keep the vigil as the Hebrews did in the Passover. The vigil represents 'those who endure until the end shall be saved.'

Paul wasn't talking about the Passover feast, but the original Passover that the feast was meant to commemorate. It refers to the last of the ten plagues of Egypt, just before the Exodus. God told Moses that the tenth plague would involve the death of the firstborn son of every household throughout the land of Egypt, and how the Israelites could be spared this fate:

Exodus 12: 1-13:

1 And the LORD spake unto Moses and Aaron in the land of Egypt saying,

2 This month shall be unto you the beginning of months: it shall be the first month of the year to you.

3 Speak ye unto all the congregation of Israel, saying, In the tenth day of this month they shall take to them every man a lamb, according to the house of their fathers, a lamb for an house:

4 And if the household be too little for the lamb, let him and his neighbour next unto his house take it according to the number of the souls; every man according to his eating shall make your count for the lamb.

5 Your lamb shall be without blemish, a male of the first year: ye shall take it out from the sheep, or from the goats:

6 And ye shall keep it up until the fourteenth day of the same month: and the whole assembly of the congregation of Israel shall kill it in the evening.

7 And they shall take of the blood, and strike it on the two side posts and on the upper door post of the houses, wherein they shall eat it.

8 And they shall eat the flesh in that night, roast with fire, and unleavened bread; and with bitter herbs they shall eat it.

9 Eat not of it raw, nor sodden at all with water, but roast with fire; his head with his legs, and with the purtenance thereof.

10 And ye shall let nothing of it remain until the morning; and that which remaineth of it until the morning ye shall burn with fire.

11 And thus shall ye eat it; with your loins girded, your shoes on your feet, and your staff in your hand; and ye shall eat it in haste: it is the LORD's passover.

12 For I will pass through the land of Egypt this night, and will smite all the firstborn in the land of Egypt, both man and beast; and against all the gods of Egypt I will execute judgment: I am the LORD.

13 And the blood shall be to you for a token upon the houses where ye are: and when I see the blood, I will pass over you, and the plague shall not be upon you to destroy you, when I smite the land of Egypt.

And everyone who had enough faith to listen to God's prophet and obey the commandment that was given to them was saved from the coming destruction that was to befall everyone else in the land.

In the 1 Corinthians passage, Paul says that "Christ our passover is sacrificed for us." This is an explicit parallel to John the Baptist calling Christ "the lamb of God." Christ was the sacrificial offering, and the parallelism is clear. His blood, and our faith and obedience to God's commandments, provides a way for us to be saved from the spiritual destruction that will otherwise come upon all the world, just as the blood of the Paschal lamb, and their faith and obedience to God's instructions through Moses, allowed the Israelites to be saved from the physical destruction that would otherwise come upon all the land of Egypt.