There are many aspects to this question, but I would first deal with the most direct one and leave the rest for later comments and feedback.
Christ died because of our sin. Therefore, first we have to know what sin is. Sin is not just doing "bad" things like stealing, lying, or even killing. The core of sin is not knowing God as God. We all have sinned in that we all are born into this world not immediately knowing who God is, and not immediately acknowledging Him. All of us, at one point in our lives, were separate from God. This is a result of the original sin from Adam. But the sin also lies on us as well.
The result of our sin, is our death. Not just physical death, but death as separation. That is, once we are separated from God, we remain always separated. The problem is that God does not have sin, and therefore there is separation. On the other hand, God does not think that this separation is totally good. It is better for us to be with Him. So He worked out a way for us to be with Him again, even though we have sinned. The answer is Jesus.
Jesus Christ is God. Although He is God, He took on our sin, and died. This at once forms a bridge. It is both shocking as well as beautiful all at once, because God who cannot even have a hint of sin, took on our sin. And God, in whom is life itself, died. This might sound impossible, or ridiculous, or contradictory at first. But consider that there is a lot about God we don't fully understand. Furthermore, what is impossible to man, is possible to God. To live and die at the same time, to be pure but tainted at the same time. This is a radical concept, that perhaps we can slightly identify with in the form of our own emotions. How we can feel so completely elated but hollow, sad and happy, worried but at peace, angry yet in love, hate but respectful. How we can be many persons yet the same person at one time. Is it any more impossible for God to have this characteristic too but in infinite terms?
In any case, once Christ died, we see that the separation is made null, because God crossed over to our side, and in doing so, brought us back to Him. There is identification and acknowledgment. There is also a form of payment, as it were. A ransom for the price of sin. Sin meant death for us, but in our stead, Christ died. Ironically, it is Christ/God that defined that sin means death. So in essence, He was both fulfilling His own law as well as abolishing Himself and His law.
The story doesn't end there of course, because Christ also rose again from the dead.