Don't just believe this because it's a popular thing in New Age teachings, if you just believe it you'll really miss the deeper understandings that come through personal revelation. You can come to this realization through your own experience, it's quite simple to see for yourself the illusionary nature of fear. When one of your fears arises, don't run away, don't feed into it, don't resist it, don't fight it, just observe it. You strengthen your fears by avoiding them, and that's all fear is, avoidance.
What happens when you really look at your fears without backing down? They lose some of their power over you, don't they? When you really really look at them, feel what's behind them, look at the fear entirely, it disappears! It was never real to begin with, but you gave it an illusionary existence by feeding the fear with your life energy. Fear is really nothing more than this, a parasitic thought form.
Compare what happens when you look at fear to what happens when you look at Love. Put your attention on your own Heart, your Connection to God. When you pay attention to these loving feelings, when you really focus upon your own Heart, the Love grows, doesn't it? Fear is fed by avoidance, while Love is fed through Awareness. Fear and Love, unconsciousness and Consciousness, avoidance and acceptance, illusion and Reality. Which is your choice? I recommend paying more attention to the way you behave than what you tell yourself in your mind. Watch for patterns that arise, experience that keep repeating themselves.