Jesus died on the cross to make atonement [reconciliation/amends/reparation] for us because of our sins. Sin is a charge made against mankind because we were disobedient to the [commandment/law/character/will] of God. And the wages of our sin is [spiritual/moral/physical] death. Sin is what separates us from fellowship and communication with God because God is holy and cannot stand to look upon, fellowship or communicate with sin, and in this case, sinful humanity. When Satan tricked Adam and Eve to sin, they no longer possessed God’s Holy Spirit inside of them, because God does not dwell within sinful vessels. The Holy Spirit, by God’s grace, mercy and love for humanity, still walked with them though, because they believed and had faith in God still and wanted to be obedient and do what was pleasing in God’s sight. Noah, Abraham, Moses, King David, Peter, Paul and etc. also were sinners, who fell into sin occasionally in their walk, but they continued to walk with God via a repentive heart and by faith. Not everyone in the Bible had this same heart though. Pharaoh, Judas, etc. did not want to walk with God.
When Jesus came to earth and died on the cross, he took the charge of our sins upon himself, and the full wrath of God, that was against humans, were instead put on Jesus. That’s why when on the cross, “And about the ninth hour Jesus cried out with a loud voice, saying, ‘Eli, Eli, lama sabachthani?’ that is, ‘My God, My God, why have You forsaken Me?’ ” (Matthew 27:45-46)
God could not even look upon his own Son while he was on the cross. Their fellowship was broken at that moment for the first time in eternity and God’s Spirit left Jesus. That is because, though Christ was with God in heaven since the beginning, though he was perfect, holy, flawless and knew no sin, while on the cross, he became sin for us. Every sin of the world and the charge that sin brought [death and separation from communication with God], Christ became those sins and took upon himself that charge while he was up there. Then he died a death that we deserved. He died in our place so that we could live. Then, because death could not hold him down, he resurrected from the dead 3 days later so that we could have that same hope in Him for a resurrection of our dead sinful state in order to have a newness in life now, and a chance to go to heaven in the time to come.
When we repent for willfully living in our sins, and believe in Jesus and put our faith in him as our lord and savior, our charge is now swapped with him. We put our sins on Him/the cross, and he takes off his righteousness and places it on us, “covering us with his blood,” so that we will become forgiven. Because his blood was pure, and because we as believers are now spiritually covered in his blood, when God sees us, he doesnt see sin, all he sees is the blood of his perfect son on us. Therefore, though we may still physically sin here on out until death [because we are still in this corrupt body], we are now justified in Christ because of the blood and will not be charged with death or condemnation, but rather complete sanctification in our bodies and eternal life once God comes back to restore his kingdom.
But being justified does not mean that we continue in our sins on purpose though. God’s grace for us does not give us a license to sin. That is why you must understand the price it cost for God to send his son to die for us. When we understand the love God the Father must have had for us in order to sacrifice his perfect and only and only begotten son, for sinful, undeserving humanity, our remorse for our evil deeds and our love for him in return restrains us from wanting to commit willful sin. It causes us to turn away from acting in sin in order to be obedient to His will again. This is called true repentance. Sin is what made Christ have to die for us in the first place. So we should develop a mindset of, “since Christ died for me, the least I could do is live for him.” And once we fully understand and believe this gospel, God will send his Holy Spirit to be with us and in us again, to give us power over our sinful desires, to help convict us and keep us from sin. This is why you hear so many Christians give their “testimonies” or accounts as to the life they used to live before accepting Christ, and their life now after they accepted him. [Nope, they still arent perfect, but they are living and striving for perfection by denying their sinful desires in order to submit to God’s will].
Forgiveness and redemption is only for those who receive and apply the blood though, not for everyone else in the world. That is why John 3:16 says, “For God so loved the world, that he sent his only begotten son, that whosoever shall believe in Him, shall not perish, but have everlasting life.” Only those who so chooses to believe will be saved from the charge of sin.
So when someone says Jesus died for our sins, it’s meant that he died for the charge of our sins [death], so that anyone who would believe and receive him will not be charged with sin or be condemned. Sin will be here and in us all until Christ comes back, but we as believers, must die daily to our sins and rest on the promise in Christ that he will completely cleanse us from it once he comes back.