Early evidence (and personal experience) so far tells me that this deal is highly-targeted (may or may not work for you) and tends to have most success if you have an AMEX "Blue" or Platinum card. Read more ...
HERE is more info on the deal.
The deal worked on some of my Blue and Platinum cards (my wife's "authorized user" card - but not my primary card!), but so far haven't heard of it working on any other AMEX cards (such as Costco, Delta Airlines, etc.).
But, IF IT WORKS, it is a good/quick/easy $15 of free stuff from Amazon.com! So worth a check.
If you aren't familiar with AMEX Offers, I won't spend too much time talking about it here (I should do a separate/full post about this sometime soon!). But you can probably Google information about it. In a nutshell, you just log-in to your online AMEX account and click on the tab for "Offers" - that you can load onto your card. They are usually things like $10 off $50 at a certain merchant (or in this case $15 of $15+ at Amazon!), in the form of a statement credit (so you buy something for a certain minimum amount and then you get a, usually immediate, statement credit on your AMEX statement).
Advanced move: If you have multiple AMEX cards, you can either have multiple AMEX logins or one login, with multiple tabs (one for each card on your account). Either way, when you are "claiming" these AMEX offers, it is best to open a different browser window (one in Internet Explorer, one in Chrome, one in Firefox, and/or maybe "incognito" versions of the browsers) for each of your accounts (or individual card tabs within the account). Then, if multiple cards have the same offer, you can "claim" then on each card. Otherwise, SOMETIMES, the offers can only be claimed on one card and then they are not active to be claimed on other cards. Hope that make sense, if not, I'll expand on the info later and/or you're always welcome to ask me. (Or HERE is a more detailed description on how to make sure you get multiple versions of the same offer on different cards in your name.)