In a nutshell, the New Evangelization is a fulfillment of the mission Christ gave the Twelve Apostles to "go and make disciples of all nations", but it focuses specifically on those who have already heard some part of the Gospel and have either lost it (the seed was choked out by weeds), never become rooted in it (the seed was sown on rocky ground), or had it snatched away before they could truly receive it (the seed was eaten by birds). The New Evangelization uses "new" methods that are actually straight from the Bible.
First, we as evangelizers have to find out where the people we talk to are. Just as St. Paul in Athens began his testimony by speaking to the Greeks about the true nature of the "unknown god" they already worshiped, we also meet the youth and young adults right where they are. The most common difficulty we have encountered in our work with teens isn't that they don't know enough of what the Church teaches (though to be sure, this is frequently the case); it's that they don't even think there's a reason to consider that God exists! We have to come to them in their doubts and show them that faith truly does affect our lives before they will even listen to what we have to say about God.
Second, we can't just force students to accept the truth. If we simply stood up and read them the Catechism, many would not understand it, and even those who did would not accept most of what we told them. Jesus recognized this, and preached in parables. Even to His closest followers, much of what He said was clear only after the Crucifixion and the Resurrection. Like Jesus, we speak in terms that are understood and do not try to force students to learn what they aren't yet ready to comprehend. (That said, we are always faithful to the Magisterium of the Church and are careful to not give the impression that we believe in heresies when the truth is hard to understand.)
Third, we don't merely evangelize. We teach others to spread the Good News just as we have given it to them. St. Paul and the Apostles did this, fulfilling the Great Commission, and it is because of them that we have a Church today! We refer to this as discipleship, and it is one of the primary goals of our bi-weekly Bible studies (Inspire! and Teen Disciples).
I hope this post gives you a better understanding of the New Evangelization and why it is so critically important to the future of the Church. But also remember that the most important thing you can do for us as a SPIRITUS team and for the Church as a whole is to keep praying!
-Jared