Even if an app it free, you still have to have a credit card attached to your account. If that's not the issue, make sure that your Fire is linked to your Amazon account. Barring that you'd probably need to contact Amazon.com tech support directly for troubleshooting assistance.
It is not downloading, because may be you have not filled in your credit card information. You will have to do that even if the app is free. Your money will be safe until you buy an application or any other thing.