It makes sense, really. Think of it mathematically (oh, boy, an English nerd doing math...yes, it happens sometimes). If x+x=2x, you just have more of the same garbage that was there before. x+-x=0. The garbage is canceled out by its opposite. :)
It is this concept that allows the lives of the Saints to be so profound.
It is also the concept behind much of what happens in the works of Tolkien, especially as regards his treatment of heroism and self-sacrifice. In Lord of the Rings, Sauron's defeat is made possible by Frodo and Sam's self-giving. Tolkien believed that Evil should be given minimal attention. Characters who study it too deeply, even if their original intent is to defeat it, often are enthralled by it, and become minions of the Enemy. Evil, Tolkien suggests through this, should be given just enough attention to make its defeat possible, and the best way to defeat it, is to keep one's eyes on Good.
The fellowship survives because they respond to the presence of evil by being strong for the side of good and Truth, supporting each other even in the face of mortal danger. This does not rule out the possibility of armed battle, but it does mean they approach conflict with a different attitude: one of protecting and building what deserves to be loved, rather than utterly destroying out of anger or envy. The result of this is twofold. First, Evil is ultimately defeated. Second, because they focus their attention on protecting good, the fellowship remain unconquered by the machinations of their Enemy, and their souls (including that of Boromir, who sacrifices his life to protect others), come through the War of the Ring safely, their goodness preserved.
The Holy Father is really much more articulate on the details of how this plays out in the world as we know it. Here's a bit of what he says:
This Gospel passage is rightly considered the magna carta of Christian non-violence. It does not consist in succumbing to evil, as a false interpretation of "turning the other cheek" (cf. Lk 6: 29) claims, but in responding to evil with good (cf. Rom 12: 17-21) and thereby breaking the chain of injustice. One then understands that for Christians, non-violence is not merely tactical behaviour but a person's way of being, the attitude of one who is so convinced of God's love and power that he is not afraid to tackle evil with the weapons of love and truth alone. Love of one's enemy constitutes the nucleus of the "Christian revolution", a revolution not based on strategies of economic, political or media power: the revolution of love, a love that does not rely ultimately on human resources but is a gift of God which is obtained by trusting solely and unreservedly in his merciful goodness. Here is the newness of the Gospel which silently changes the world! Here is the heroism of the "lowly" who believe in God's love and spread it, even at the cost of their lives.
Read the rest here.