Read more about Fear from the works of the Mother and Sri Aurobindo.
What is Fear
Fear, desire and sorrow are diseases of the mind; born of its sense of division and limitation.  Fear is a phenomenon of unconsciousness. It is a kind of anguish that comes from ignorance. One does not know the nature of a certain thing, does not know its effect or what will happen, does not know the consequences of one's acts, one does not know so many things; and this ignorance brings fear. One fears what one does not know.  Fear is also a terribly contagious collective thing—it is much more contagious than the most contagious of illnesses. 
Fear is manifested in different parts of selves i.e. physical fear, mental fear and vital fear. In majority of the people, physical is governed by the subconscious and fears that are driven out of active consciousness take refuge in that subconscious. Thus, the physical fear is in the subconscious hidden by the mental and vital fear which are much more conscious. 
Why there is Fear
Fear stems from a sense of inferiority. 
There are three reasons. First, an excessive concern about one's security. Next, what one does not know always gives an uneasy feeling which is translated in the consciousness by fear. And above all, one doesn't have the habit of a spontaneous trust in the Divine. 
Because in the large majority of men, the body receives its inspirations from the subconscient, it is under the influence of the subconscient. All the fears driven out from the active consciousness go and take refuge there and then, naturally, they have to be chased out from the subconscient and uprooted from there. 
How to Overcome Fears
If you remain vigilant, then with the increase of the Force upholding you, a power of self-control will come, a power to see and reject the wrong turn or the wrong reaction when it comes. Fear and unhappiness will not give you that. It is only by this vigilance accompanied by an opening to the supporting and guiding Force that it will come. 
One of the great remedies for conquering fear is to face boldly what one fears. You are put face to face with the danger you fear and you fear it no longer. The fear disappears. 
In order to set out on these paths without fear and without any danger, one must have organised his being with the help of reason around the highest centre he consciously possesses, and organised it in such a way that it is inwardly in his control and he has not to say at every moment, "Ah! I have done this, I don't know why. Ah! That's happened to me, I don't know why"—and always it is "I don't know, I don't know, I don't know", and as long as it is like that, the path is somewhat dangerous. 
And the truth is that if instead of being shut up in the narrow limits of their little person, they could so widen their consciousness as to be able not only to identify themselves with others in their narrow limits, but to come out of these limits, pass beyond, spread out everywhere, unite with the one Consciousness and become all things, then, at that moment the narrow limits will vanish, but not before. 
Fear and Religion
The fear of the Gods is the beginning of religion.  Fear of the gods arose from man's ignorance of God and his ignorance of the laws that govern the world. The idea of the divine in the form of judge, king exaggerates the importance of the sense of sin and thereby prolongs and increases the soul's fear and self-distrust and weakness. The fear of god is useful only when the soul has not yet grown sufficiently to follow good for its own sake and needs an authority above it whose judgment it can fear and found upon that fear its fidelity to virtue.
Courage is the total absence of fear in any form.  True courage, in its deepest sense, is to be able to face everything, everything in life, from the smallest things to the greatest, from material things to things of the spirit, without a shudder, without physically ….without the heart beginning to beat faster, without the nerves trembling, without the slightest emotion in any part of the being. Face everything with a constant awareness of the Divine Presence, a total self-giving to the Divine, and the whole being unified in this will; then you can go forward in life and face anything whatever.  When the vital is entirely surrendered to the divine it is full of courage, enthusiasm. 
Everyone possesses two opposite tendencies of character, in almost equal proportions, which are like the light and the shadow of the same thing.  Thus, when one has fear, one also has courage. And to get over this impurity of fear one has the resources in the form of courage, tapas and surrender.
Fear is a hidden consent. When one is afraid of something, it means that you admit its possibility and thus strengthen its hand. It can be said that it is a subconscient consent. Fear can be overcome in many ways. The ways of courage, reason, faith, knowledge, surrender are some of them.