The paradox of thrift
Matt Yglesias offers a clear, non-technical explanation. I would add a few points:
1. If wages are relatively flexible in the downwards direction, it is easier to avoid the downward spiral from falling aggregate demand. There is an odd tension (though not contradiction) between the view that a stimulus is necessary and the Progressive view that workers don't have much bargaining power and that bargaining power really matters.
2. Savings are especially likely to fail to translate into investment when the banking system is messed up. That applies today, but Keynes was not sufficiently aware of the importance of this condition.
3. Prior to the collapse, savings out of income in this country were approximately zero. So the notion of "people ending up saving less" has to mean a rising ratio of debt to gdp. That's OK for the argument, but now it gets complicated.
For instance, instead of "saving more" the core action under consideration might be to pay down some debt instead of spending money on consumption. But what does the creditor do with those funds? Are dollars sent to creditors "low velocity dollars" rather than "high velocity dollars"? Maybe, but of course they don't have to be. If the citizenry is paying back to creditors who engage in active lending (or for that matter rapid consumption), and make new loans rapidly, things can be OK. If the citizenry is paying money back to zombie banks, maybe those banks just sit on the cash. (How much of the money is going to zombie banks?)
A lot of claims about the paradox of thrift depend on having a good handle on which micro-sectors of the economy breed high vs. low velocities of money. We don't always have such information. The whole notion of how money can get trapped in "low velocity circuit," beyond simple observations about first-round effects (the poor spend a higher percentage of their incomes than the rich), is receiving insufficient attention.
The "Treasury view" that you can treat monetary velocity as constant is wrong. But there's a lot about monetary velocity which we don't understand, or at least which we have not yet applied to the current problems at hand.