From Wikipedia:
if a certain number of people sign up for the offer, then the deal becomes available to all; if the predetermined minimum is not met, no one gets the deal that day. This reduces risk for retailers, who can treat the coupons as quantity discounts as well as sales promotion tools. Groupon makes money by keeping approximately half the money the customer pays for the coupon.
For example, an $80 massage could be purchased by the consumer for $40 through Groupon, and then Groupon and the retailer would split the $40. That is, the retailer gives a massage valued at $80 and gets approximately $20 from Groupon for it (under a 50%/50% split). The consumer gets the massage, in this example, from the retailer for which they have paid $40 to Groupon.
Unlike classified advertising, the merchant does not pay any upfront cost to participate: Groupon collects personal information from willing consumers and then contacts only those consumers, primarily by daily email, who may possibly be interested in a particular product or service
So basically the business agrees a discount with the web site, the web site markets the deal and takes payment upfront, and then passes 50% of the revenue to the business which then supplies the goods or services to the customers. By doing it that way the web site gets the money up front from (say) 200 or even 1000 customers per deal, which is a pretty good business model. It benefits the businesses by paying them a guaranteed amount for a greatly increased number of customers, so it's a quantity discount as far as they are concerned.