Tap the button in the invitation email, then Join the list, signed in with the invited address; if that address has an account, Accept in Updates works too.
Invitations don't expire, so there's no rush. Invited to a collection? It works the same way, with Join the collection.
Steps
From the invitation email:
Tap the "Join the list" button ("Open the list" if you're invited as a Viewer).
The page shows who invited you and, under "What you'll be able to do", what your access allows; "What can each access level do?" has more. Tap Join the list.
Not signed in? The page says "You'll need to sign in with the address the invitation went to." No account on it yet? See "How do I sign up or sign in?".
You join as soon as you're signed in, and the list opens.
No email? If the invited address has an account:
Sign in with that address.
Open Updates (see "How do I find my way around?") and find the invitation under Waiting for you. It reads "[name] invited you to “[list]”".
Tap Accept. Home shows it too.
Don't want to join? Tap No thanks on the invitation page or in Updates (without an account, it asks you to sign in first), or ignore the email: nothing changes.
What if it says the invitation is for a different account?
"That invitation is for a different account." means you're signed in with another email: tap Switch account, sign in with the invited address, then tap Join the list. The page shows which account you're in; switching signs you out first. Signing in with Google or Facebook? Pick the account with that address. Can't get into it? Ask the owner to invite the address you do use.
What if it says there's nothing waiting, or I declined by mistake?
Sign in with the address the invitation went to; if you declined by mistake, ask the owner to invite you again. An answered invitation shows "There's nothing waiting at this link." to anyone not signed in with that address, even you. Signed in, you'll see the list if you'd already joined, or "You turned this one down." if you declined. Still nothing waiting? The owner may have taken it back, so ask for a new one. "That list isn't there any more." means it was deleted.
Where do I find lists shared with me?
Lists you've joined are in Wishes → Lists, marked Shared; with two or more lists, switch the All lists menu to Shared with you to see only those. Only got a link, not an invitation? Then the list is on Home: "Where can I see the gifts I've reserved?" says where.
How do I leave a list shared with me?
Tap the Shared with you badge under the list's title, then Leave wishlist in the "Leave this wishlist?" dialog. To rejoin, the owner has to invite you again. On a collection, it's Leave collection.
Tips
Open the invitation where you're signed in with the invited address, and joining takes one tap.
A forwarded invitation only works for the address it went to, so anyone else needs their own from the owner.
Lots of lists? In Wishes → Lists, switch Recently opened to By name; it sticks.