In this description of Jay Gatsby Nick is referring to the idea of the Platonic ideal, a philosophical construct presented by Plato in Ancient Greece.  According to this theory every object and every being has an ideal representation, a “perfect” iteration, if you will, upon which its form in this world is based.  For example, if you consider a series of desk chairs, they are based on a similar design and yet remain different, remain flawed in different ways.  They are functional but they are not a perfect ideal.  Therefore there must exist an ideal in some abstract world, otherwise we would have nothing upon which to base the structure and function of these real, flawed chairs.

Likewise with concepts such as love, truth, despair…the ways in which we experience these things are mere copies of their ideal forms, and as such subject to idiosyncrasy.  With the creation of all things in this tangible world we naturally strive toward as genuine a copy of these ideals as possible.  So, when Gatsby is described as having “[sprung] from his Platonic conception of himself,” it implies that he is trying to force his real self into the ideal image of himself.  Moreover, he does not consider this ideal an abstract of the man Jay Gatsby that exists in some neutral, unattainable realm of ideas, but a fully attainable version of himself that he has, himself, created.  He has an imagined ideal that he is trying to conform to by attempting to make himself, in a word, perfect, in the eyes of Daisy and the upper crust of society. 

This indicates that Gatsby is trying to suppress the real-world, flawed persona of his past and become a wholly new character, created out of this abstract ideal to walk the earth in West Egg.

This phrase comes from Chapter Six of The Great Gatsby as Nick Carraway describes the concealed boundary of Gatsby's vision of his life which he creates himself out his own imagination ("Platonic conception of himself"), a vision in which, in his confusion of values and romance, he is "faithful to the end."

This vision which has "sprung" from the imagination of Jay Gatsby is one of gaudy materialism that is the American Dream of the Jazz Age. For, as he lay in his meager room by the south shore of Lake Superior as a youth, the conception of his desires began to form as an outlet for his imagination which was dulled by the tedious labor in which he was engaged at St. Olaf college in southern Minnesota where he worked as a janitor. Then, after returning to the shores of Lake Superior where he meets Dan Cody and begins working for him, "the vague contour of Jay Gatsby had filled out to the substantiality of a man."

You are referring to the part of the novel where Nick says:

"Jay Gatsby sprang from his Platonic conception of himself"

Plato was a Greek philosopher who, among many other things, was concerned with the real world versus the ideal world. So when Nick says this, he means that in a sense, Jay Gatsby invented himself. He invented an ideal of himself and then tried to live up to that ideal. He was not happy with his "real self" (a poor man from an uneducated family), so he re-invented himself as an educated and wealthy man.

In Greek mythology, it is believed that the Godess Athena "sprang full grown from the head of Zeuss" so the use of the word "sprang" is also a reference to the Greeks, only mythology in this sense. The Jay Gatsby that appeared on the stage of East and West Egg "sprang" from his own creation, but Nick soon learns who the real Gatsby is when he meets Gatsby's father.

Read about it here on eNotes.