It won't... Kindles use Android
Adobe stopped ALL development of Flash for Android last year. (July 2012)
There are older versions of Flash Player (for Android) at that link, but there are no guarantees any of them will work.
See also: http://forums.adobe.com/thread/1061194
I've read a lot about "Dolphin" for Android https://play.google.com/store/apps/details?id=mobi.mgeek.TunnyBrowser
I'm not an Android user (no tablet or phone) so I can't make any endorsements that would hold any credibility, but most of the Flash solutions I've read point to Dolphin as the best alternative.
Check the Android forums: http://androidforums.com/ to see what their "gurus" recommend.
The latest Kindle Fire HD is not officially supported by Flash Player. Please see this page for a list of all certified devices:
That said, some users have been able to get Flash Player working on the Kindle Fire by using Dolphin, a third party browser.