- God is Holy
- God is Just
- God is Merciful
The combination of these attributes is the answer to your question.
- His Holiness demands that He be obeyed, unequivocally.
- His justice demands that He deal with disobedience, mercilessly.
- His mercy demands that He forgive, lovingly.
How does one strike a balance in a meaningful manner? The answer is that He who knew no sin became sin, and God poured His wrath upon His Son. Unraveling the Biblical language a little bit, this means that our (the elect's) sins were imputed to Christ, Christ's righteousness is imputed to us, and Christ was no longer considered perfect but sinful. Because He took our sin upon Him, we are now blameless before God, but Christ was not. Christ took the merciless wrath due us, satisfied God's justice for those under this covenant of grace, and permits God to show mercy to us because God's justice has been satisfied in Christ. The key here is that Christ became sin by taking our sin (and the corollary, for another question topic, is that we obtained Christ's righteousness). Only God has the power to do this.
The work on the cross was more than physical: the penalty Christ paid was much more than just physical death and suffering. I recommend this one-hour video for a more encompassing explanation from a Baptist/Puritan viewpoint.
Once you understand these points (obedience, justice, mercy), the rest makes perfect sense. There are deeper questions to be asked at this point (and not on this same SE question!) but I think you are better off to think of them on your own at this point.
Edit, to Address more questions from the O.P:
The transfer of the sin was not the purpose of Christ's death on the cross. It was part of the penalty that He paid for taking our sin upon Himself. In the beginning, man disobeyed God (Genesis 3). This man is named Adam. Adam is the head representative of the human race. Because he sinned, his nature changed and he was lost in disobedience without the ability to obey God. Because Adam, who was flesh, sinned, his flesh also became corrupt and he eventually died.
As I said, the point of Christ's work on the cross was to consume the wrath of God for those whom the work will cover (I know this is a confusing way to word it. I'm trying to avoid opening another topic entirely. If you have a question regarding who the work was intended to cover, it is worth another question entirely). This moral and physical death is called the curse of Adam. Christ's death on the cross satisfied the physical side of the curse of Adam. Christ's being forsook by His Father satisfied the spiritual aspect of the curse. The physical death took place in order that He would identify with the flesh and conquer sin in the flesh.
To use the term "murder" is inappropriate, here. Murder implies malice and an innocent victim. When Christ took our sin, He was no longer innocent and His death was not in malice, it was in justice.
Once God imputed our sins to Christ, if God would not have punished Him, His Holiness would have been denigrated because His justice would not be satisfied, as if to say: "Yeah, it's okay if you disobey me. No biggie." The distinction is that it is never ok to disobey God and simple "forgive and forget" could not work, or else God would be giving up His Holiness. The most heinous act you can conceive of doing to a fellow man is not as heinous as the slightest insult you can think of against God (pretending that there is a way to wrong a man without wronging God in the process).
Paul Washer's point in the video at 6:20 was that because Christ is God's Son, all of creation and events in time are for His glory, because the Father loves the Son. If we are obeying God because we want to avoid punishment or to "be moral," then we are doing it in the wrong spirit. All things are to glorify Christ.