Open the wishlist link the owner sent you, find the wish you want to give, and use its Buy link to go to the store. You buy it there. Wishpicks doesn't sell the item, so you pay the shop and arrange delivery with them.
Reserve the wish first, so nobody else buys the same thing.
Steps
Open the wishlist link. You don't need an account to view it.
Find the wish you'd like to give.
Click Reserve on that wish and confirm. That marks it as taken, so a second person can't reserve it.
Click the wish's Buy link to open the store.
Buy the item there, paying and choosing delivery as you normally would.
The Buy link only hands you off to the shop the owner saved. Wishpicks never handles the payment. See "Does Wishpicks process payments?" In some regions your first click also brings up a one-time card ("You're almost at [store]") asking whether to allow affiliate links. Allow and continue and Continue without both take you to the same store at the same price.
Good to know
Reserving isn't buying. It marks the item as taken so two people don't turn up with the same gift. See "How do I reserve a gift before buying?"
Each item can be reserved by one person. If a wish is already reserved, someone else has it, so pick a different one.
The surprise holds. By default the owner doesn't see that anything was reserved, and your name stays hidden. See "Can I see who reserved or bought a gift?" in Gift Reservation.
There's no address field anywhere on Wishpicks. Arrange delivery with the recipient yourself, or ask them for their details privately. See "How will someone know where to send my gift?"