It's simple accounting, as interest payments on the debt become an increasingly higher part of the budget, the end result becomes insolvency.

Of course, an actual default for a country like the U.S. is a very unlikely outcome for a variety of reasons. What is more likely, is that inflation is allowed to rise and 'eat' up the debt. The "hope" is that a mild rate of inflation is sufficient to do so. I doubt it though, there have always been periods of high inflation, and there is no reason to think they will not return.