When we say that Jesus Christ died for us, we simply mean that we deserved to die, but Jesus died in our place. The Bible says, "But God demonstrates his own love for us in this: While we were still sinners, Christ died for us" (Romans 5:8).
Let me explain it this way. Suppose you had committed a serious crime and were brought before a judge for sentencing. There was no question about your guilt; you knew you had committed the crime, and you knew you deserved the sentence the judge was about to hand down. You had broken the law; the penalty had to be paid - and you were the one who had to pay it. Then the judge pronounced the sentence, and the deputies started to take you away to jail.
But then, suppose the judge suddenly ordered them to stop, stepped down from his bench and announced that he was going to take your place! You deserved to go to prison - but he was going instead! In other words, he was innocent but was willing to take upon himself the penalty that the law demanded for your crime - the penalty you should have paid.
This is what Jesus Christ has done for us. We deserve to die and be separated from God forever because of our sins - but Christ took our place. By His death on the cross He paid the price for our sins - fully and completely. In other words, He died for us; He died in our place. Now all we have to do is accept His forgiveness, by trusting Him alone for our salvation. Have you opened your heart and life to Christ?