You buy the product in the online store the wish links to, not on Wishpicks. Every wish carries a Buy link that opens the shop 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 that 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; it doesn't sell products or take your payment. See "Does Wishpicks process payments?" Owners add wishes by pasting a product link from any online shop, so the Buy link could lead almost anywhere.
Good to know
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. Either answer gets you to the same store at the same price.
Reserve the wish before you head off, so no one else buys the same gift. See "How do I reserve a gift before buying?"
Stores change prices and sell out, so a link can go stale. If the product looks unavailable or different from the wish, check with the owner before buying.
There's no address field on a wishlist. Arrange delivery with the recipient directly, or message the owner privately if you need their details.