It's easy to see why extending Unemployment Insurance benefits will raise aggregate demand. People on UI are typically in marginal economic situations and are going to have a high marginal propensity to spend. If you own a store in a town with a lot of unemployed people, you're definitely going to see sales go down if people's benefits are cut off. That's going to have downstream impacts on your suppliers and your staff. But it also seems intuitive that a government program which, by definition, you become eligible for by not finding a job will have an averse supply-side impact on the labor market. Yet here, courtesy of Mike Konczal, is an empirical look at the state of play:
As you can see, there's no clear indication that UI eligible people really are staying out of the labor market in a substantial way. One can understand that in part in terms of the fact that there are substantial non-pecuniary reasons to not be unemployed (respect of your peers, etc.) and those particularly apply to the previously employed. All things considered, I'm sure you could imagine a better-designed program that currently existing Unemployment Insurance but the bottom line is that in addition to important humanitarian benefits it does a fair amount to stimulate demand and very little to restrict labor supply.