The idea is not that you can spend your way out of debt, that would be silly. The idea is that you can and should spend money to compensate for lack of demand in a recession, because the markets overshoot. Once you are out of recession, you must pay back the debt.
So austerity when the economy is good, spending when it is bad. Anti-cyclical government behavior.
Austerity in a recession is pro-cyclical. As was predicted and Greece (for example) has shown, a government cannot save itself out of debt in a recession, because tanking the economy at a crucial time like that makes the debt worse (as percent of GDP, and that's the crucial number in terms of ability to repay).
Update:
I am German, "living within your means" is sort of part of my DNA, and not living within your means is going to cause problems. But timing does matter!