They don't sell the house. According to the article, they cannot sell it at a price high enough to satisfy the lienholders. Rather than shouldering the entire burden of a bad property investment alone, they can force the bank to take some of it.

If you have negative equity in a property that has negative cash flow, you really should get rid of it as fast as humanly possible, by any means at your disposal, no matter how much beans and rice you typically eat in a month.

Definitely not cheaper than no house payment. In NV for example there are thousands of homedebtors who haven't made a payment since 2008. They're effectively protected by state law until 2017.