Your Kindle Fire uses the Android Operating System.
Adobe stopped ALL development of Flash for Android last year. (July 2012) That includes Kindle as well as Samsung Tablets.
This is from Samsung Support (07/31/2013):
"Beginning August 15, 2012 we will use the configuration settings in the Google Play Store to limit continued access to Flash Player updates to only those devices that have Flash Player already installed. Devices that do not have Flash Player already installed are increasingly likely to be incompatible with Flash Player and will no longer be able to install it from the Google Play Store after August 15th, 2012."
And from Kindle Support (07/31/2013):
"The Amazon Silk web browser on Kindle Fire 1st Generation supports some, but not all, versions of Flash. Adobe Flash is pre-installed on a 1st Generation Kindle Fire and is disabled by default.
If you see a message to install Flash while using your Kindle Fire, this means that the content you're attempting to view uses Flash. If you want to view the content, you'll need to enable Flash through the Silk settings menu.
If a web page suffers from slow performance due to Flash, or you're unable to view a web page with Flash enabled, we recommend switching to mobile view when browsing a Flash-heavy website to see if that solves your problem:
Some popular websites may also have an app available. Check Amazon Appstore, accessible through the Apps library on your Kindle Fire, for availability.
The Amazon Silk web browser on Kindle Fire 2nd Generation, Kindle Fire HD 7", and Kindle Fire HD 8.9" doesn't support Flash.
Some sites use alternative web technologies to Flash in their mobile versions. We recommend switching to mobile view when browsing a website with Flash content to see if that solves your problem:
Some popular websites may also have an app available. Check Amazon Appstore, accessible through the Apps or Games library on your Kindle Fire, for availability."
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.
I had the same issue. If you're on YouTube, scroll down to the end of the page. At the bottom where it says "terms, privacy,...." there is also a link for mobile version. Clicked on it and that worked for me....
I tried clicking on the menu of my kindle fire, request another view, mobile" but that didn't work for me...