How does Jesus' blood save us from our sins?

Okay, so, growing up in church, I heard many times in many ways that Jesus died on the cross, and that that saves us from our sin.  But I never understood how that worked out exactly.  I think I even remember asking someone and them replying, "it just does."  O...kay?  I had this question lingering in my head until one day, I was reading the footnotes in 1 John in my Life Application Study Bible, and it said, 

" How does Jesus' blood purify us from every sin?  In Old Testament times, believers symbolically transferred their sins to an animal, which they then sacrificed (see a description of this ceremony in Leviticus 4).  The animal died in their place to pay for their sin and to allow them to continue living in God's favor.  God graciously forgave them because of their faith in him, and because they obeyed his commandments concerning the sacrifice.  Those sacrifices aniticipated the day when Christ would completely remove sin.  Real cleansing from sin came with Jesus, the "Lamb of God, who takes away the sin of the world" (John 1:29).  Sin, by its very nature, brings death - that is a fact as certian as the law of gravity.  Jesus did not die for his own sins; he had none.  Instead, he died for the sins of the world.  When we commit our lives to Christ and thus identify ourselves with him, his death becomes ours.  He has paid the penalty for our sins, and his blood has purified us.  Just as Christ rose from the grave, we rise to a new life of fellowship with him (Romans 6:4)."

Ohhh....