How can the terms redeem and coupon be used together?

Redeem in the English language means to buy back. The coupon is a piece of paper that has a set value for money as applied to a particular product. If one buys that particular product, the store will buy back the coupon. That is, the store will give you the value of the coupon off the purchase price. In some cases, the coupon is worth as much as one half of the total price of the item, therfore you get the product for half the regular price. Americans do not as a rule barter for anything.