I'd always assumed that Republicans made this argument as a matter of simple rhetorical dishonesty; Democrat = democracy, Republican = republic. If the United States was founded as a republic only, Republicans -- by virtue of their party's name -- could convince the weak-minded and logic-challenged that Republicans were closer to the founders' original vision. Not the best or most rational argument, but -- let's face it -- the GOP doesn't waste a lot of time on outreach to brainiacs. Think back to the Tea Party protests for examples of the deep thinkers the party attracts.
But there's a darker reason for the argument; Republicans aren't big fans of democracy. Here's the co-founder of the rightwing Heritage Institute and the Moral Majority, the late Paul Weyrich:
Now many of our Christians have what I call the goo-goo syndrome -- good government. They want everybody to vote. I don't want everybody to vote. Elections are not won by a majority of people, they never have been from the beginning of our country and they are not now. As a matter of fact, our leverage in the elections quite candidly goes up as the voting populace goes down...[CLICK TO READ FULL POST]