As you can see my individual point falls below the equilibrium, so I do not purchase any Asian food. However, if the price were lower, I would buy some Asian food. By using a service such as groupon, it effectively lowers the price of the food. For my example I was able to get a $20 coupon for only $10. This shifted out my demand curve by $10 (because my $10 could now buy $20 worth of food).
You can see that I am now at the equilibrium point where I will decide to eat some of their food. This result is identical to the economics of traditional advertising (tv, radio, newspapers, etc.) in that it shifts the demand curve right (an increase in demand). However, the reason it shifts out is different. With traditional advertising we see an increase in demand because because of either an increase in information or a change in "tastes or preferences". These cause the demand curve to shift right because people desire more of the good because their attitudes have changes.
With respect to these coupon sites, the curve shifts right because we have changed the price of the good for consumers. Rather than spending a certain amount on a meal, they are spending less with the help of the coupons. So if I was only willing to spend $5 on the good, but I can buy a $10 coupon for $5, then in effect I am willing to "buy" $10 worth of the product (with the coupon).
But how does the coupon company benefit from these deals? They charge a percentage to the company, so not only does the company have to you the consumer a big discount, but they have to give the coupon company a cut as well. So why do they do it? Most of the companies with coupons offered on groupon are new or seasonal. If they need to jump start their business, or survive through a slow period then they are willing to give massive discounts to get customers. There hope is that you will love their service and become a repeat customer. If you really like their service maybe you will even send some friends their way.
***This post uses groupon as an example for this economic concept, however, in order to be transparent I do get paid if you click their ad and buy something