You buy the product in the external online store that the wish links to — not on Wishpicks. Each wish carries a Buy link that opens the store the owner saved it from, and you complete the purchase there.
Steps
Open the wishlist and find the wish you want to give.
Click the wish's Buy link.
You'll land on the product page in the external store.
Buy it there, paying and arranging delivery as you normally would on that site.
Wishpicks isn't a shop. It saves gift ideas and links out to stores, without selling products or taking your payment — see "Does Wishpicks process payments?" And since owners add wishes by pasting a product link from any online shop, the Buy link could lead almost anywhere.
Good to know
You might see a quick consent card first. In some regions, the first time you click Buy you'll see a one-time card ("You're almost at [store]") about allowing affiliate links. Whichever you choose, you reach the same store at the same price.
Reserve first. Before you head to the store, reserve the wish so no one else buys the same gift. See "How do I reserve a gift before buying?"
Stores sometimes change prices or sell out, so a link can fall out of date. If the product looks unavailable or different from the wish, check with the owner before buying.
Wishpicks has no delivery-address feature, so there's no address on the wishlist — you buy at the store and arrange delivery directly with the recipient. If you need address details, message the owner privately.