This idea was first elucidated in the Fable of the Bees by Mandeville in 1714, and later popularized by John Maynard Keynes in the twentieth century. I remember catching on to this in a intro to business thought class at the University of Utah. We were reading Thoreau (yes, the class was awesome) and it dawned on me that if everyone lived like he did the economy would grind to a halt. Despite it being a paradox, this makes sense, right? If everyone stopped spending money at the same time, whether to save the money or to pay down debt (as is currently happening), then much of the demand for goods and services (economic growth relies on), just wouldn’t be there.
Wikipedia defines the paradox of thrift thusly:
[I]f everyone tries to save more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population because of the decrease in consumption and economic growth. [emphasis theirs]
So, as governments around the world cut their spending their economies will grow more slowly. This is because the paradox not only applies to spending by individuals, but also that done by companies and governments as well (because it doesn’t really matter who is doing the spending). Two other points seem appropriate here. First, a country’s debt is considered in terms of the size of the country’s economy (i.e., debt/GDP). If you slow down the growth of GDP by slashing spending, your debt to GDP ratio won’t fall even if you’re slowing the growth of your debt. Lower tax receipts from the slowing economy are another big part of the problem. Rather than slashing spending to the bone, for governments a better way to fix the problem is to grow their way out of debt, as the US did after WWII. In this situation not only are tax receipts up, but the GDP (i.e., the denominator in the ratio) is increasing. This is the problem with Italy right now; it’s not growing, and the austerity measures only make things worse.
The second point I wanted to make is that currently the US government can borrow money at negative real interest rates for up to seven years. This means that the government can borrow money now, and, after inflation, pay less back in the future. No one really mentions this when discussing the country’s budget problems. With the crumbling infrastructure in this country, and the ~24 million people currently un or under-employed, it would make a lot of sense to borrow the money now, build stuff we need by paying currently-idle people to do work, and then pay the smaller amount of money back in the future. With negative real interest rates, this literally has no downside. It's unfortunate Europe's not so lucky.