Why does Fitzgerald reveal the truth about Gatsby's background at this particular point in The Great Gatsby, Chapter 6?
Fitzgerald places Gatsby's background story in Chapter 6 because he weaves the true story of Gatsby, one "James Gatz of North Dakota," into Nick's discussion of the "[c]ontemporary legends" that surround Gatsby, such as the legend that Gatsby's house is really a boat that moves "secretly up and down the Long Island shore." Fitzgerald is establishing the unique nature of Gatsby's character. He does so from several perspectives to create a perception of objective opinion thus freeing opinion from Nick's private, subjective perception.
As stated, one way Fitzgerald does this is to talk about the contemporary legends. Another is to introduce momentarily a young reporter eager for "a comment" from Gatsby:
About this time an ambitious young reporter from New York arrived one morning at Gatsby's door and asked him if he had anything to say. ... "Why,--any statement to give out."
Yet another is to introduce the truth of Gatsby's background. However, there is more than this to the placement of this background information; it serves a deeper purpose in characterization.
At the end of Chapter 5, we have just witnessed the tour of Gatsby's house with Nick and Daisy.
We went upstairs, through period bedrooms swathed in rose and lavender silk and vivid with new flowers, through dressing rooms and poolrooms, and bathrooms with sunken baths-
The last we see is that Daisy says "something low in his ear" to Gatsby and that he responds by "turning toward her with a rush of emotion." Nick prefaces his departure from the scene by explaining that the "feverish warmth" of Daisy's voice was, for Gatsby, a "deathless song" that held him captive in a dream that could not be "overdreamed."
His hand took hold of [Daisy's] and as she said something low in his ear he turned toward her with a rush of emotion. I think that voice held him most with its fluctuating, feverish warmth because it couldn't be over-dreamed--that voice was a deathless song.
After this, when we learn Gatsby's story, we can answer for ourselves some of the questions we have about why he is so intent about Daisy, what happened to make him as he is, and what he is, really, beneath the legends, the news, and the facade.