It should be said, that though belief in "tax cuts will pay for themselves" type theories is frequently expressed by Republican Party politicians and Republican-oriented columnists, talk show hosts, and think tankers, I do think it's unlikely that this theory actually causes Republicans to advocate for tax cuts. For that, you need to look to something more like Jason Furman's testimony today (PDF) before the Ways and Means Committee.
Here's a chart that summarizes his main conclusions and, I think, basically explains Republican thinking. If the image is too small for you to read, click and you'll get a bigger one:
What we're seeing here are dynamic analyses of the effect of tax cuts on the income of people at different income levels, under different scenarios. The top selection shows what happens if you pay for tax cuts via decreases in spending -- i.e., what conservatives typically say should be done. Most households -- 74 percent of them, in fact -- wind up worse under this scenario than they would have been without the tax cuts. But 57 percent of families in the top twenty percent benefit. And 99 percent of families in the top one percent benefit. If, by contrast, the tax cuts are paid for simply by a future re-raising of taxes, then 76 percent of households wind up worse off, but 43 percent of households in the top one percent benefit.
In the real world, of course, you're likely to see a mix of spending cuts and tax increases used to close deficits (so indicate the precedents of the major deficit reduction packages of the past) so that Republican Party advocacy of tax cutting does, according to Furman's analysis, probably achieve its purpose of further enriching the richest Americans, albeit at the expense of the vast majority of Americans. For some of the movers behind this policy, it's a question of self-interest (they're rich or being paid by the rich to serve their ends) and for others it's a simple question of justice -- a not entirely trivial minority of the population regards the fact that the very richest Americans aren't even richer as deeply immoral.