For those who had trouble with the Kindle Fire, you can actually just download the file from Handster and open it within your browser. From the Home screen, go to the Quick Settings icon and select More. Go to Device. Then select ON for "Allow Installation of Applications". Then go to the Handster link, download the APK file and open it from your downloads (like you would anything else you've downloaded on your kindle) and bam, installation is open.
If you already downloaded it, and don't know how to retrieve it you can get the "ES File Explorer" app from Amazon which allows to browse your directories. The apk file will be in your Downloads folder. Just open it from that browser and the installation will begin.
If you want to add this information to your original post it may be helpful to any Kindle Fire users since it is a little harder for us to use third party apps. It seems most of us have figured it out but sharing information is always good.. Either way, great app. I will be donating.