How does cutting unemployment benefits help the economy?

I understand that letting the unemployment benefits expire will reduce government spending (and therefore the budget deficit) in the short term, but won’t reducing/eliminating income for millions of people result in reduced consumer spending and cause more problems like more jobs lost due to lack of consumer demand than it solves in the long term?

And if many of the people who are no longer collecting unemployment wind up on food stamps, how much money is the government really saving?