How does Jesus' death and resurrection save humanity?

This question is to be answered by examining marriage law and how it provides the framework for both collective and individual salvation. The Mosaic Covenant was essentially a marriage between Jehovah and His "chosen people" Israel. Moses was the instrument in Jehovah's hands who performed the labor of salvation to bring a remnant out of bondage in Egypt and to enable them to become a sovereign nation in the land of promise that Jehovah promised Abraham would be the land where his posterity would be blessed. As is always the case, covenants have conditions. And, unfortunately, Israel was not a faithful "bride" to Jehovah. In due time Israel essentially committed adultery against Jehovah by putting aside the statutes and judgment He ordained for Her and by adopting the corrupted ways of Her neighbors. As a consequence for this adultery, Jehovah had both Israel and the adulterous men who enticed her "stoned to death". This can be seen as the siege and destruction of Samaria by Assyria first and then subsequently the siege and destruction of Jerusalem by Babylon. And, subsequent to doing their dirty work against Israel and Jerusalem, both Assyria and Babylon were also destroyed. This is the consequence of adultery. Israel became scattered and mixed in among the "Gentiles" as if She was the bits and pieces of a deceased and decomposing corpse with only bones remaining scattered about. While in this state Israel would be subject to many privations and sufferings because She was no longer under the system of governance and protection that She formerly enjoyed. She would not have the ability to rise above the corrupt powers around Her and She would remain in this state until the terms and conditions of the Mosaic Covenant were all fulfilled, every last bit of it. This includes the punitive conditions. Thus, it has been a bit of a mixed blessing and cursing to the Gentiles to have Israel mixed and diffused in among them. They bring the potential of their future blessings and redemption as well as the consequences of their violation of the Mosaic Covenant.

So, if this basic approach is understood, this context also provides the reason someone like Jesus was required to die in order to "save" mankind. The terms in which salvation can be understood is all about being an heir of the Father's Kingdom. Much is spoken about how when the Father's Kingdom comes that those who inherit it are heirs of salvation. This essentially means that there shall come a day when God will establish a new covenant with a new "chosen people" and this people will be purged and refined and ultimately enabled to receive a new "land of promise" where they will be under oath and covenant to have integrity to the statutes and ordinances the Father's Kingdom requires. This is, essentially, a new marriage. However, there's a problem if God is going to marry Israel again. God's marriage law forbids a husband from receiving back a wife that he has formerly puts away. Once she is put away she causes whatever man with whom she marries to commit adultery also with her. There is only one thing that releases a woman from what is called the "law of her husband" and that is if he dies. Once this aspect of Israelite marriage law us clearly understood and it is applied on the level of God and His Bride Israel, the conclusion you naturally come to is in order for Jehovah to establish a new covenant with Israel, He first must die. Once Jehovah dies, Israel is then freed from the restriction and becomes eligible to be remarried to Jehovah. This is also why it is so important that Jehovah be raised from the dead. Otherwise, what would be the point?

So, what I am getting at here is this: Jesus became Jehovah incarnate by receiving within himself the spirit of Jehovah and provided official representation such that on Jehovah's behalf Jesus could suffer death in order to break the bands that prevented Him from being able to receive Israel into a new covenant.

In order to answer your question, about how Jesus death on the Cross can and does pay for the sins of all men; we must take a look at how we got into this mess to begin with and what God demanded in repentance for sin. These verses from Genesis give us the reasons that is possible. Gen 2:9 And out of the ground made the LORD God to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of knowledge of good and evil. Gen 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul. Gen 2:16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat: Gen 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. There are two very important points that we must remember; 1. God breathed the breath of life into Man (Adam), but God nowhere breathed life into any other human being. That includes Eve who was made from Adam's rib, this is because life already existed in Adam's rib. 2. God told man (not just Adam alone) that that life would be demanded for his disobedience. Now comes the essential, but hardest part to comprehend' Ready? Everyone from Adam to the newest newborn shares that same breath of life God breathed into the first man. That includes the man Jesus, (but not the Deity Jesus which was alive in the beginning). That was the LIFE that Jesus sacrificed on the Cross! That fulfilled God's demand of life in reparation for SIN.

Salvation comes when we allow our life inherited from Adam to be also sacrificed on the same cross with Jesus. We do this when we ask Jesus to live within us, as Paul said in; Gal_2:20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me. This ETRNAL LIFE, is given to us by Christ at Salvation; Joh_17:2 As thou hast given him power over all flesh, that he should give eternal life to as many as thou hast given him.

Some other Scriptures which may help you to understand that sacrifice are; Mat_10:39 He that findeth his life shall lose it: and he that loseth his life for my sake shall find it. Mat_16:25 For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it. Mat_19:17 And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments. Mat_20:28 Even as the Son of man came not to be ministered unto, but to minister, and to give his life a ransom for many. Mat_25:46 And these shall go away into everlasting punishment: but the righteous into life eternal. Mar_10:30 But he shall receive an hundredfold now in this time, houses, and brethren, and sisters, and mothers, and children, and lands, with persecutions; and in the world to come eternal life. Joh_3:36 He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him. Joh_5:24 Verily, verily, I say unto you, He that heareth my word, and believeth on him that sent me, hath everlasting life, and shall not come into condemnation; but is passed from death unto life. Joh_6:27 Labour not for the meat which perisheth, but for that meat which endureth unto everlasting life, which the Son of man shall give unto you: for him hath God the Father sealed. Joh_10:17 Therefore doth my Father love me, because I lay down my life, that I might take it again. Joh_17:3 And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent. Rom_6:6 Knowing this, that our old man is crucified with him, that the body of sin might be destroyed, that henceforth we should not serve sin.

There are verses I have read that have escaped me for now that speak to what I am about to say. Sin and corruption are destructive naturally and spiritually. The sacrifice that was Christ... was pointed to frequently in the OT, and at the precise time "Christ died for the ungodly" in a reality that I see little to no discussion about in the sense of an actual experiential realty. Roman's 5:6 For while we were still weak, at the right time Christ died for the ungodly. The reality that I speak to is not a parable, or something mystical or hard to understand. The reality is that without the cross, sin that had fed into the underlying fabric of our existence over the years was set to erupt into the visual world, and mankind would have been destroyed. The sacrifice of Christ, the spotless lamb - stopped that... dead in it's tracks. At the time of Noah, sin had also run it's course, and it did exactly what was it's purpose - total destruction.. other than God's provision for continuance of humanity through Noah. It is evident that history is repeating itself, but this time there will be no boat (other than the Rapture possibility), and no second sacrifice. Christ will come as a King. The scriptures speak him coming as a warrior to set things right. As for me, I do all I can not to lean on any thing I have, but to whole lean on Jesus name.

Romans 6:23 “For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord”. This is a spiritual law, it is as certain as any natural law.

So, essentially, we understand that death is separation. In this case, the death spoken of above is separation from God.

Jesus always makes manifest the will of God the Father, yet he was put to death. Christ (who never sinned) suffered death in every way, even unto separation from God the Father. Christ died for sin as the law requires, however, it was not for His own sin, but for the sins of mankind.

Mathematically, this spiritual law is like this: A(life) + B(sin) = C(death)

In Christ it becomes: C(death) – B(sin) = A(life)

Sin being separateness from God.

Bear with me. Three part answer.

Part 1:

Jesus replaced the system of animal sacrifice. Here is a nice video showing and explaining animal sacrifice: https://youtu.be/pGfdjEzaKtg?t=2m22s

Why move away from animal sacrifice? It doesn't fix the underlying issue. There will be a need for constant sacrifices to keep up with the sinful nature of humans.

Like the lamb in the video which doesn't have spots or blemishes, Jesus is Holy and clean (sinless). When He died on the cross he took all our sins on Himself, if you so choose to give your sins over to Him.

Part 2:

Jesus who is one with our heavenly Father, He is God represented in flesh (John 1:14). The enemy plotted against Jesus to have Him crucified (Matthew 26). This is orchestrated through the sins of man e.g. Judas giving in to greed to betray Jesus etc. etc. While Jesus is on the cross He shouts out "Father forgive them for They know not what they do" (luke 23:34) (our sins are being thrown on to Him and he is being "separated" from the Father (Matthew 27:46)). Jesus passes and three days later he raises from the dead. This shows that all that sin that caused him to separate from the Father was not enough to kill Him (He conquered it).

Part 3:

"For wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord" Romans 6:23.

When you decide to accept Jesus into your life he does a work in and through you. The Holy spirit guides you to become more like Christ, to be become one with Him. If you decide to get baptised it represents you dying to your old self when you go under the water (you join Jesus in the tomb where he was buried) and when you rise you are a new creation in Jesus (Romans 6:4). You are on a journey to become one with Christ and all your faults are being covered by the blood that Jesus spilled ("the crucifixion He underwent").

When the Father looks at someone who has accepted Christ into their life He sees the righteousness of Jesus Christ when He looks at you. (Ephesians 2:5-6 ; Romans 10:9).

this is a good question, in the old Testament an animal sacrifice was required to be given to God as an offering to atone for the sins that were committed after the last sacrifice,

I have borrowed the first paragraph from Why did God require animal sacrifices in the Old Testament? a website I found with a quick search on animal sacrifice in the bible.

God required animal sacrifices to provide a temporary covering of sins and to foreshadow the perfect and complete sacrifice of Jesus Christ (Leviticus 4:35, 5:10). Animal sacrifice is an important theme found throughout Scripture because “without the shedding of blood there is no forgiveness” (Hebrews 9:22). When Adam and Eve sinned, animals were killed by God to provide clothing for them (Genesis 3:21). Cain and Abel brought sacrifices to the Lord. Cain's was unacceptable because he brought fruit, while Abel's was acceptable because it was the “firstborn of his flock” (Genesis 4:4-5). After the flood receded, Noah sacrificed animals to God (Genesis 8:20-21).

when Jesus came He was the perfect sacrifice, He was unblemished, as was required of the animal sacrifices. the innocence of the animal covered our "not being innocent"

another Paragraph from the site listed above says it very well.

Why, then, do we no longer offer animal sacrifices today? Animal sacrifices have ended because Jesus Christ was the ultimate and perfect sacrifice. John the Baptist recognized this when he saw Jesus coming to be baptized and said, “Look, the lamb of God who takes away the sin of the world!” (John 1:29). You may be asking yourself, why animals? What did they do wrong? That is the point—since the animals did no wrong, they died in place of the one performing the sacrifice. Jesus Christ also did no wrong but willingly gave Himself to die for the sins of mankind (1 Timothy 2:6). Jesus Christ took our sin upon Himself and died in our place. As 2 Corinthians 5:21 says, “God made him [Jesus] who had no sin to be sin for us, so that in him we might become the righteousness of God.” Through faith in what Jesus Christ accomplished on the cross, we can receive forgiveness.

Read more: http://www.gotquestions.org/animal-sacrifices.html#ixzz2fHn1afk6

Additional

the Animal had to be YOUR Animal, your BEST Animal, it was a sacrifice that you made, something that you gave up because you wanted to atone for your Sin. God made a Sacrifice by giving us His Only Son to be the Ultimate Sacrifice.

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,

1. The wages of sin is death (Romans 6:23)
2. All have sinned (Romans 3:23)
3. 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.

1. It's the blood of Christ which can wash away our sins and make us perfect like Jesus Christ.
2. It's the resurrection of Christ which gives us victory over death.
3. It's the wounds of Christ which give us healing.
4. 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".