The basic "balance sheet recession" (Koo 2011) story begins with a debt-fuelled asset price bubble, for example:

Japan housing 1992
US housing 2007

The main argument is that when the bubble bursts the value of people’s assets collapses, but the value of their liabilities remain. Their balance sheets are “under water”.

In this situation, people need to engage in balance sheet repair. This involves private sector deleveraging (increase savings, pay off debt); or firms trying to reduce debt rather than maximise profit. Collectively, this reduces AD and generates a prolonged slump.

The problem is that the central bank can’t do much, for three reasons:

People don’t want to borrow because they are focused on balance sheet repair (therefore low interest rates aren’t enticing)
People draw down bank deposits to pay debt (money supply contracts and the money multiplier becomes 0)
Lenders themselves (i.e. banks) have their own balance sheet problems

Reinhart and Rogoff provided empirical support, showing that recoveries following financial crises are inherently weaker. However this has been challenged by Nelson and Lopez-Salido (2009):

“We find that the regularity that recoveries are systematically slower in the aftermath of financial crises does not hold for the postwar United States. The pace of the expansion after recessions seems to reflect deliberate aggregate demand policy. A weak lending outlook does not appear to pose an insurmountable obstacle to the functioning of stimulative aggregate demand policies”

Scott Sumner provides an alternative (simple) explanation to balance sheet recessions:

Recession caused by tight money
Tight money reduces nominal income (one might ask why recipients of debt repayments don’t spend it, but this implies there’s an excess money demand problem)
Since most debts are nominal, this implies bigger declines in spending in more highly indebted areas

In other words, weak NGDP growth explains things perfectly well

“the weak economic recovery is a failure of policy to fully restore aggregate demand, nothing more” (David Beckworth)
“increase in government deficits may introduce the uncertainty that causes deleveraging to occur” (Vuk Vukovic)

And even Allan Meltzer (1995, p.67)

A further reason to doubt the importance of bank lending as an independent channel propagating the Great Depression is that the decline in bank lending can be readily explained as a response to the decline in nominal GDP… there is no need for any separate explanation of the decline in bank lending

To conclude, we should factor in the structural problems at the onset of the crisis (i.e. not simply an AD shock out of nowhere) and the regime uncertainty caused by big players (government and central bank).

Koo, Richard, C., 2011, “The world in balance sheet recession: causes, cure and politics” Real-World Economics Review, Issue 58