This offer is for those with the Kindle Wi-Fi with Special Offers or Kindle 3G with Special Offers only (presumably; if anyone out there has the Kindle 3G with Special Offers, can you check your offer page and see if this one is there? I may have to buy one, if the offers are different). View the Special Offers on your Kindle (Menu, Special Offers) and you should see a new promotion (if you don't, turn on your wireless, click Menu, Sync & Check for New Items, then after the sync has completed, go to the Special Offers screen):
Spend $5 on Kindle Singles and get a $5 Amazon.com Gift Card
Click on offer, then click on the link on the offer page sign up and get an email with the promotion code. Sign-up for this offer expires on June 2.
You'll get an email (right away), with a link to Kindle Singles page. However, there are no promotional codes to enter, you just have to click to have the offer sent to your email (for this particular offer, I'm not sure that you even need to do that, but I would, just in case). Then, purchase Kindle Singles adding up to at least $5 between March 29 (today) and June 15 (midnight PT). The $5 gift card will automatically be applied to your account sometime in the month after that (no later than July 15). Full details are here.
I already know the first single I'm going to get: Bittman's Kitchen: What I Grill and Why ($1.99), by Mark Bittman. I've already looked at the sample and was considering buying it, so this deal means I really only have to purchase an additional $3.01 in Kindle Singles to get the gift card.
With the way most singles are priced, that essentially means I have to buy two more at $1.99, since I would end up five pennies short if I bought a $0.99 Single and one at $1.99 (or three at $0.99). Without finding one that ends in .49, you'll actually have to spend $5.95 to get the $5 gift certificate, with this particular deal. So, if you were considering buying a few (or find a few over the next month that interest you), this deal lowers their price. It doesn't, though, give you a 100% return on the money spent (or more, as the first American Express deal did).