How does Jesus' death and resurrection save humanity? Time and again we hear that Christ died for our sins, or that Christ died so that we could be saved. But I've never understood how exactly the two are connected? In what way does the death of Christ benefit us, or lead to our salvation?

Update

This may be asking too much, but what I'm really looking for is the rationalization (essentially, God's reasoning) behind it. @Mawia and @Malachi have both offered very good and very thorough answers, but neither really addresses what I'm really looking for (most likely because I haven't explained that well enough).

The answers have provided great references to scripture to show that sinless blood is consistently required to atone for sins, and that man is in a perpetual state of sin, therefore atonement is required. But I get stuck on the first part: why does someone without sin need to be punished in order to atone for another person's sins? I understand Christ sacrificed Himself, He chose to pay for our sins, and that's important because it shows His love for all of us. But it doesn't explain why sacrificing something that is sinless is even acceptable, let alone a suitable payment for sins.

Just to answer @PeterTurner's point: I'm open to any answer from a learned Christian perspective: the important thing to me is that it explains some reasoning behind it, even if that reasoning is not universal among Christians.
The atonement theology is a complicated one to explain. To the believers, it is a simple acceptance of one's imperfection and the sacrifice of Jesus Christ the perfect one to render us perfect in the eyes of God. To the unbelievers, it is foolishness that God would become human to die for their sins.


  For the message of the cross is foolishness to those who are
  perishing, but to us who are being saved it is the power of God. (1
  Corinthians 1:18, NIV)


Some facts from the Bible,


The wages of sin is death (Romans 6:23)
All have sinned (Romans 3:23)
All must die because all have sinned (Romans 5:12)


"Sin entered the world through one man"
Sin is like a deadly disease which infected the first man Adam and the effect continued down to all generations. There was no death before the fall of man in the garden of Eden. When Adam and Eve disobeyed God and ate the fruit from the tree in the middle of the garden, sin entered the world and death along with it, for the wages of sin is death. Man is now sinful and cannot be with God.


  Therefore, just as sin entered the world through one man, and death
  through sin, and in this way death came to all people, because all
  sinned. (Romans 5:12, NIV)


"The blood of Christ is the remedy for sins"
Man cannot create a cure from sin. Man cannot do anything to stop sinning. Only God can cure us from our sickness. The medicine to cure sin and death is the Blood of Jesus Christ the Son of God. Only the blood of Christ can cleanse us from all sins.


  But if we walk in the light, as he is in the light, we have fellowship
  with one another, and the blood of Jesus, his Son, purifies us from
  all sin. (1 John 1:7, NIV)


"Christ died for us" Jesus Christ not only poured out His blood but also gave His life. The death penalty of sin was also paid by Jesus by dying on the cross.


  But God demonstrates his own love for us in this: While we were still
  sinners, Christ died for us. (Romans 5:8, NIV)


"Christ defeated death" After receiving the death penalty, Jesus again rose from the death to declare victory over death. The resurrection of Christ enables us to share in His resurrection.


  And if Christ has not been raised, our preaching is useless and so is
  your faith. (1 Corinthians 15:14, NIV)
  
  For if we have been united with him in a death like his, we will
  certainly also be united with him in a resurrection like his. (Romans
  6:5, NIV)


"We will be resurrected like Christ" The main purpose of the death and resurrection of Christ is that, one day we will be resurrected like Christ to live forever with God and there will be no more death. God will cleanse us from all our sins by the blood of Christ and we will be transformed to immortality.


  Listen, I tell you a mystery: We will not all sleep, but we will
  all be changed— in a flash, in the twinkling of an eye, at the last
  trumpet. For the trumpet will sound, the dead will be raised
  imperishable, and we will be changed. For the perishable must
  clothe itself with the imperishable, and the mortal with immortality.
  When the perishable has been clothed with the imperishable, and the
  mortal with immortality, then the saying that is written will come
  true: “Death has been swallowed up in victory.”
   “Where, O death, is your victory?
      Where, O death, is your sting?” (1 Corinthians 15:51-55, NIV)


"Why does someone without sin need to be punished in order to atone for another person's sins?" It's not only the punishment that plays the main role here.


It's the blood of Christ which can wash away our sins and make us perfect like Jesus Christ. 
It's the resurrection of Christ which gives us victory over death.
It's the wounds of Christ which give us healing.
It's the punishments received by Christ which liberates us from the eternal punishment of Hell. 


The whole idea is, 

"You cannot save yourself from sin and death, you need God to do that for you".