Great quote. It comes very early in the book and gives some insight into Nick's character and the way that people treat Nick. Your question mentions "Nick's desire." I don't believe that Nick ever had a huge desire to know "the secret griefs of wild, unknown men." The reason that I think this is because of the sentence that immediately follows.
"Most of the confidences were unsought — frequently I have feigned sleep, preoccupation, or a hostile levity when I realized by some unmistakable sign that an intimaterevelation was quivering on the horizon; for the intimate revelations of young men, or at least the terms in which they express them, are usually plagiaristic and marred by obvious suppressions."
What Nick means by these two sentences is that for some reason eccentric and often powerful men like to confide in Nick. Nick doesn't actively seek it out, though. In fact, he says that he often intentionally avoids such situations.
Gatsby is one of those wild, unknown men, but Nick doesn't seem to mind the secret griefs of Gatsby.
"I wanted no more riotous excursions with privileged glimpses into the human heart. Only Gatsby, the man who gives his name to this book, was exempt from my reaction — Gatsby, who represented everything for which I have an unaffected scorn."
Nick isn't sure what it is about Gatsby that makes him different, but Nick does seem to believe that there is a beauty to Gatsby's personality. A sort of romanticized ideal that allowed Gatsby to be a very optimistic man in Nick's eyes.
". . . there was something gorgeous about him, some heightened sensitivity to the promises of life, as if he were related to one of those intricate machines that register earthquakes ten thousand miles away. This responsiveness had nothing to do with that flabby impressionability which is dignified under the name of the “creative temperament.”— it was an extraordinary gift for hope, a romantic readiness such as I have never found in any other person and which it is not likely I shall ever find again."
The thing that absolutely turned Nick away from knowing the secret griefs was not Gatsby himself, but what other people did to Gatsby. All of the gossip, rumors, back stabbing, two-faced attitudes, and general using of Gatsby. People were not interested in being friends with Gatsby. They were interested in what Gatsby could do for them, either financially or socially.
"No — Gatsby turned out all right at the end; it is what preyed on Gatsby, what foul dust floated in the wake of his dreams that temporarily closed out my interest in the abortive sorrows and short-winded elations of men."