When Adam and Eve sinned, God made them clothes from animal skins. That's the very first sin, and the very first time that a blood sacrifice had to be made as a result of sin.
It has been that way ever since. The Jewish people had to make a yearly sin offering; in it, a lamb was slaughtered and its blood offered as a sacrifice for the sins of the people.
It was predicted that a Messiah would come and make the sacrifice permanent. Jesus was that Messiah, and he himself was the sacrifice, his blood shed for the sin of men. As an innocent, sinless man (even the Qur'an agrees with this), he was a sufficient sacrifice. As an all-powerful God, his sacrifice was powerful enough to be propitiated to all men for all time.
At the time of Jesus' sacrifice, the very thick veil that separated men from the inner part of the Jewish temple was rent in two, from top to bottom. This signified that man was able to come into God's presence; he no longer needed a high priest. Jesus is that priest that makes intercession between us and God the Father.
Islam says that Allah is forgiving, but there are no guarantees. Furthermore, it teaches that those who are good enough go to paradise, and those who aren't are punished in fire. But that means there's an arbitrary line separating "good enough" from "not good enough." That hardly seems fair, especially to the person who needed just one more good deed to avoid torment. It's also not fair because although Allah may indeed be forgiving, he cannot be just if he does not punish wrongdoing.
What of the person who makes a mistake and in a fit of rage or even by accident does harm to another person? Let's say that someone accidentally kills a mother and their child. Can their one sinful life be enough to pay for the two lives they've taken? Such a person can NEVER pay their debt on their own. And how about the person that lives their life for themselves, acting selfishly, and then later has a change of heart? Although they are sincere in wanting to serve Allah, they'll never be able to overcome the bad they've done; they're destined to be burned. Paradise has to be earned in Islam; it's a pay-off, a wage.
God is just. He punishes all wrongdoing, and does so fairly. There is no arbitrary cut off for those who are forgiven and those who are not. There's no need to "work off" our debt of sin. We can all -- each and every one of us -- be forgiven for our sins if we simply allow Jesus to pay the penalty for them on our behalf. That's grace!