9to5mac.com's Mike Beasley reports that the new update is required before Apple users update to iOS 7
If this isn't done before the new Apple iOS 7 is installed, then, Beasley warns, "this could cause users to have to login again and re-download their books."
The Kindle app update is downloadable at the iTunes Store now.
The Apple explanation there is displayed in FAINT purple and I personally can't read it. What possesses web developers to make sure that the important text is barely seeable, much less readable? Is it just that it's an attractive color? Or, maybe, it's the idea that Apple users may be younger than most.
Details of what the app does are linked to at this Kindle Apps page.