It probably has something to do with the fear of having leaders with a "snob" element, or ones who have lost touch with the average American. Or maybe it's just jealousy. Seriously, though, what most people consider to be snob behavior has less to do with education and more to do with family prestige. I know several people attending Ivy League or other very prestigious universities, and none of them ever act the least bit smug or superior because of what college they go to. The people who act the most snobbish usually come from the families of the upper-upper class, or the "blue-bloods". Some (not all, but some) of them can be real stuck-up pricks. But simply attending a good college doesn't automatically turn you into an asshole.
Also, I find it interesting that the dumbass you quoted is talking about the GOP. In almost all cases, the Republicans are the ones who market their candidate as being "just plain folk", and label the Democratic candidate as being "elite". And the Tea Party in particular isn't exactly recruiting members from the ranks of America's most intelligent. Odd. But in any case, I would rather have a snob for a president than a layman.