I think Groupon works on four things:
1) Provides immediate or near-immediate cash flow to businesses in a bad economy who do not necessarily have great options for raising capital quickly. If you sell 3,000 groupons for an hour-long massage, you're about to get a check for close to $100,000 sometime within the next two weeks. Your employees have to do some work later, whatever, that is $100,000 that can pay the rent and keep the lights on today.
2) Breakage. It will depend on the particular offer, but some portion of Groupons will be sold but never redeemed. Free money for Groupon and the business, what isn't to like?
3) Customer acquisition: many of the companies use the Groupon as a loss-leader to get customers in the store for either upsells ("dinner is deeply discounted, wine is available at the standard prices") or establishing a recurring relationship. Seen in this light, it is just another marketing channel, except one which causes positive cash flow right after you sign on the dotted line as opposed to negative cash flow.
4) Some businesses which offer groupon have unit economics where a marginal customer is essentially pure profit, and anything they can do to get a marginal customer is economically justifiable as long as it doesn't cause spillover effects to the main business. A lot of the service industry is like this: as long as you operate below 100% capacity, the cost of servicing one additional customer is too low to measure. (Beauty salons which pay salaries pay whether the manicurist has someone's hands in hand or not.) As long as you don't cannibalize your existing customer base, it is worth getting a marginal manicure customer at nearly any price you can negotiate.