You can download the apps through your computer, or open the Amazon app on your Android device, pull out the side menu, and select "Apps & Games."
From here, just hit the banner up top and browse through the apps. When you find one to download, hit the "Free" button at the top of the page, then "Get App" to begin downloading it, and finally install it like normal (making sure you have app installation from "Unknown sources" enabled).
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Other worthwhile deals to check out: