Couple of ways. One is many of the coupons have codes that link back to the site's owner. The "click here" for coupon transmits the coupon site information to the vendor.
An other way is to use the coupons/discount to get you to look at the page. Show you other Ads (like Google AdSense), they get paid when you click.
Yet another way, is to get you to look at the site. Then try to sell you something directly.
Yet another way, get you to the site, have you register for personal coupons and sell you as a lead. That can be done without emailing you by placing special offers on you login page.
Of course all these methods CAN BE COMBINED to make even more money.
There are many ways to skin a cat. Somtimes, it helps to ask your self the qustion: Ok this is where I am, NOW WHAT can I do to make it better. DO IT than ASK AGAIN!
20+ years sales and marketing, also computer/web geek