What does Tom mean when he says "civilization's going to pieces" at the beginning of The Great Gatsby?
Well, let's see, Tom means to show that he is well-read and knowledgeable about the world, but what he actually does is prove that he is, in fact, an unintelligent, prejudiced bigot.
The quote you mention can be found at the first dinner that Nick spends with Daisy and Tom at their house in East Egg. This leg of the conversation begins when Tom violently (and inappropriately) reacts to Nick's joke about Midwestern crops. Suddenly, Tom is trying to sound learned by explaining Goddard's "The Rise of the Colored Empires." After insisting that everyone read the piece, he tries to expand by saying, "It's all scientific stuff; it's been proved" (13). (Note how Tom neglects to give the evidence of this.) Heck, I won't even mention how ditsy this conversation makes Daisy sound. Her idiocy is even worse here!
Then Tom goes into a racist rant saying that "if we don't look out the white race will be--will be utterly submerged" (13). He gives a long tirade with the convoluted theory:
This fellow has worked out the whole thing. It's up to us, who are the dominant race, to watch out or these other races will have control of things. ... This idea is that we're Nordics, ... And we've produced all the things that go to make civilization--oh, science and art, and all that." (13-14)
Ah, Tom's attempt at the evidence makes him sound totally asinine. In trying to prove his intelligence, Tom has proven just the opposite. The irony is that even little Midwestern, unreliable-narrator Nick notices Tom's stupidity here by saying, "There was something pathetic in his concentration, as if his complacency, more acute than of old, was not enough to him any more" (14).
In my opinion, this little conversation is an invaluable part of Tom's characerization because it shows him to be the "hulking brute of a man" (12) that Daisy accuses him of being. It should definitely add to the reader's dislike of Tom's character.