I’m very happy that you are seeking an answer this very important question. The complete answer is not a brief one, so please indulge me here. The understand the answer to how Jesus died for our sins, we must first discuss two things:
First, the penalty for sin is death according to the bible. St. Paul said it succinctly in Rom 6:23: For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord.
In this case, death is not merely the physical shut-down of our bodies, but eternal separation from God. The purpose for life in this light can be seen as an opportunity to choose to be with God for eternity (repent of our sin and be forgiven) or turn away from God forever (die unforgiven of our sin). Since God gives us a choice to sin or not to sin, then ultimately it is our choice to be with God in heaven or live apart from him in Hell.
Second, Jesus Christ was not just some great teacher or prophet or miracle worker, he was God Himself. Not only did he claim as much (John 14:6-11), but also proved it with his miracles (John 15:24).
Born separated from God, we are all pre-disposed towards sin, especially once we reach an age of reason and become mature enough to order our God-given free will away from doing good at all times. In other words, our sinful actions disqualify us from an eternity with God, and no good deed is good enough to make up for our sins.
This second point is not just some theological innovation made up by the Church, but the logical conclussion of Jewish beliefs which required the annual sacrifice of a lamb in the Temple to attone for a year of sins. Even for them, good deeds could not make up for our sins. Only the life taken from an umblemished creature could do that. Christ, being both sinless and eternal, becomes our final sacrifice that is sufficient for sins commited by everyone.
The need for our salvation then is freely offered by our sinless God with his own death on the cross. Having lived a sinless life, Christ offers us his own death in place of our own to assume the penalty of our actions that we deserve. Placing faith in Jesus Christ and asking him to indwell within you assures us of our spiritual salvation when we physically die. That is the promise of God, which is why we Catholics share this good news with others.
If one wants the gift of salvific assurance, then all one would have to do is take account of one’s sins, acknowledge one’s need for a savior, and pray to receive Christ the way one would receive a gift. Then one should seek out a Catholic Church and ask the priest for guidance to ensure one’s faith is properly disposed to receive Christ fully at a mass (classes are required for non-Catholics) and partake of the sacraments to shed our sinfulness while still on earth. We call this process living the life of grace.
Continue seeking answers, for Christ Himself did promise: “Ask and it will be given to you; seek and you will find; knock and the door will be opened to you.” (Matt 7:7)
Mike