The best answer to that is contained in Ayn Rand's novels. But I'll give my best summation here -
Man's nature - all men - is that of a rational being, whose mind works individually, in conjunction with reason and logic. But the key here is that his mind works individually. It is not the property of others, nor can others force him to think like them. He must come to his conclusions individually. The creations of this mind are also individually owned - they are the property of his individual mind, which created them. They belong to no other, and no other's need is a right to them.
But in order for this all to happen - for a man to use his individual mind to see and understand the world around him, to create, to build, to prosper - he must be concerned with his own self interest. The mind is a part of the self, and in order to use your individual mind, you must have an individual self. Being primarily concerned with others is to place their minds above yours - and to sacrifice that individual thought process that makes you human.
So, to be selfish is to be concerned with your own self interest. This puts your self and thus, your mind, as the most important things in your life. When they are the most important things in your life, you will be able to succeed and prosper and create and be happy, and no one can take that away from you - because it belongs to your self.
As a secondary consequence, you will also be the prime mover of humanity when you are selfish person. It is the selfish person who creates the bold, brilliant things that mark each new epoch in human history. You must be selfish to create a brand new work of art or design a great new building. You can't be concerned with what others think - because then humanity never moves forward. The progress of civilization depends on the man who creates - and to create, you must use your individual mind - and to use your individual mind, you must be proud of your self. You must be selfish.
Seriously, though, if you want a better explanation, read Ayn Rand's books.