'he died for our sins' doesn't it mean the Original Sin? the one why the Jews had to follow their current laws and why Christians don't follow these laws?
Original sin was the sin committed by Adam and Eve in the Garden of Eden, when they first turned their backs on God. This created a fundamental rupture between God and humanity and deeply marred human nature. All human beings after Adam and Eve are therefore born on the human side of this rift, with a marred nature: a tendency towards sin. So we say that human beings are born with, or in, original sin. All human beings are born with original sin. Personal sin is the sin that is unique to each individual because of the choices they make with their own free will.
God knew that in order to fix our now flawed human nature and to heal the damaged relationship between Himself and humanity, that humans would have to learn the hard way. So he chose a particular people, the Jews, as a starting place from which to reach out to the whole human race, and a particular bloodline within the Jews to be the forebears of Christ, the Redeemer, who would fix—redeem—the breach between God and Man and give us the opportunity to heal our broken nature.
The laws that the Jews followed in the old testament come from two specific grievous sins they committed as a people: first, after God brought them out of slavery in Egypt to be a free people, His own nation, they turned their backs on him, fashioned a golden calf, and worshipped it, and returned to the iniquitous lifestyle that the Egyptians practiced. After this incident, God gave them the laws found in Leviticus, and they repented of their sin. But the next generation sinned again, in a similar way, after experiencing even more of God’s miracles, and worshipped the false god Baal. After this transgression came the Mosaic law, found in Deuteronomy, often called the Deuteronomic law. This law did not come word for word from God, but was strongly mediated by Moses, hence “Mosaic law.” These are the laws that were specific to the Jews in their time and place, to help them avoid idolatry and the attractive, sinful lifestyles of the pagan nations around them, and which Christians are not obligated to follow.
In the old testament, sacrifice is a major theme: when the people transgress God’s laws, (and they do that all the time,) the priests offer “sin offerings” to God, as a symbol of their sorrow and repentance. When Jesus Christ, true God and true man, died on the cross for us, because he is God, his sacrifice was more than a symbol: it creates a bridge over that fundamental rupture between God and man (this is redemption) that human beings can now, with God’s grace, cross over, and offered us an efficacious—a truly saving—way to be reunited with God after we inevitably commit our own individual, personal sins (when we choose this, that is salvation.)
This is why infants are baptized: Baptism is the parents offering their child to God, and God reaching out and bringing their flawed nature back into communion with His own. And that is why we have the sacrament of Confession, or Reconciliation: so that after we commit our personal sins, because that tendency towards sin remains in our nature and we use our free will for evil, we can choose to return to Christ and say “yes” once again to his offer of salvation.
tl;dr Christ’s sacrifice on the cross redeemed the world by enabling human beings to be reunited with him, and offers human beings salvation, by which we continually strive to follow Him and leave our personal sins behind. Two separate but intimately related consequences of Christ’s crucifixion. The Levitical and Deuteronomic laws that the Jews were obligated to follow before Christ came were the result of specific sins that they committed as a nation in their history.