In chapter 3, what is the reason for Nick's breaking the story at this point?
Nick does not "[break] the story," as you say; the narrative merely shifts from one world -- that of the Buchanans -- to a party at Gatsby's house.
Gatsby lives near Nick, in "less fashionable" West Egg in a home that the latter describes as "a colossal affair," an "imitation of some Hotel de Ville in Normandy." It is a site of gaiety and excess. For Gatsby's weekend parties, aquaplanes fly onto the Sound, "his Rolls-Royce became an omnibus," and a staff of "eight servants" toiled all day the Monday after "repairing the ravages of the night before."
Interestingly, this world of hedonistic excess is disrupted by the actual presence of Gatsby. He is introduced uneventfully: "We [Nick Carraway and Jordan Baker] were sitting at a table with a man of about my age and a rowdy girl, who gave way upon the slightest provocation to uncontrollable laughter." Carraway absorbs the girl's mirthful sensibility and begins to enjoy himself -- to enjoy the excess ("I had taken two fingerbowls of champagne") and to fool himself into thinking it was all something "significant, elemental, and profound."
Suddenly there is a "lull in the entertainment," and, at this moment, the unknown man speaks: "Weren't you in the Third Division during the war?" The discussion of the First World War disrupts the glee -- and reminds us of the ways in which people use alcohol and parties to distract themselves from the memory. Gatsby and Carraway, it seems, cannot avoid it. Gatsby's question takes Carraway out of the scene before him and back to "some wet, gray little villages in France."
Arguably, this juxtaposition of memories of war with the hedonistic excesses of the Twenties is, along with questions about Gatsby's identity, a key point of the chapter. The questions around Gatsby, including Jordan's suspicion that he did not really go to Oxford and Nick's reservations about such a young man having so much money, parallel questions about national identity after the war. They also reflect the sense, expressed by Nick at the end of the chapter in his comment about being "one of the few honest people" he has "ever known," that identity and truth have become tricky concepts in the modern age.
Fixed ideas about identity, reflected by Tom Buchanan's tirade about race in the first chapter, are slipping away. The Buchanans reflect an old guard, while Gatsby reflects the newer possibilities in America: that of a poor Midwestern boy (possibly of Jewish origin, given that his actual name is "Gatz") becoming "great," which, in the context of the United States in the Twenties, was inextricably tied to his money.
Themes in this chapter -- the First World War, the equation of money with happiness, and the murkiness of identity -- question the old values while providing no answers on what the new values are.
There are a couple of things that could be pointed out as the cause, but it is difficult to know exactly why without perhaps being able to consult with Fitzgerald himself, who apparently made all the decisions about the narrative! In some ways you might also argue that Nick hasn't entirely broken the story at this point.
But he does begin to realize that Gatsby isn't everything he is cracked up to be, given that he remembers him from the war and previously and knows that his background could be fleshed out but isn't. Gatsby allows the rumors to fly.
Perhaps the reason why Nick begins to flesh things out is that he cannot wait much longer if there is to be a dramatic build up until everyone else finds out the whole story. It also helps to set up the reasoning for why Daisy won't actually totally fall for Gatsby as there has to be a connection to his less than high-class past for that to become the strong theme that it does.