Amazon Prime videos can also be watched offline on iPhone, iPad or Android after downloading them. This will get an awesome experience of enjoying the entraining videos from Amazon prime. But somehow this does not always work properly for some devices.
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To overcome this problem you have to make sure that you have Amazon Prime Membership. Moreover, you cannot download videos with household share option. Also, make sure that you are in America and you need devices like Fire Tablet (not Ist gen), Fire Phone, Android Phone, or Tablet or iPad or I phone for better downloading of videos on Amazon Prime.
If the above tips did not help then switch off and restart your device. If this also does not work then uninstall and reinstall Amazon Prime Video App.