Open the wishlist link, click Reserve on the wish you plan to give, and confirm. That marks the item as taken so no one else buys the same thing. It doesn't buy it for you, though. You still order it yourself at the store afterwards.
Steps
Open the wishlist link the owner shared with you.
Find the wish you plan to give.
Click Reserve on that wish.
Confirm.
The wish now shows as reserved, and nobody else can reserve it until you cancel. Think of it as a note saying "I've got this one." Buying is a separate step, covered in "How do I buy a gift from someone's wishlist?" A reservation doesn't complete the wish either. Once you've bought the gift, only the owner can mark it as received. See "What should I do after buying the gift?"
Good to know
Each item can be reserved by one person at a time. Try to reserve something that's already taken and Wishpicks tells you someone else has it, so pick a different wish.
Whether you need an account depends on the owner's settings. Some wishlists let anyone reserve without signing in, some ask you to sign in first, and on some, reserving is switched off.
If you reserved as a guest and then created an account in the same browser, those reservations carry over to it. Ones you made on another device or in a different browser won't.
Changed your mind? Cancel your own reservation and the item frees up for someone else. See "How to cancel a reservation" in Gift Reservation.
By default the owner doesn't see that anything was reserved, and your name stays hidden unless their settings show it. The surprise is safe. See "Can I see who reserved or bought a gift?" in Gift Reservation.