Download ES File Explorer 3 on the store. Then, go to the place where you download the ZIP file (most likely Downloads) then long press the ZIP and then click "More" and select "Extract to." Click "Choose path" then select /sdcard/games/com.mojang/minecraftWorlds. Select "OK" and you're done! If the map doesn't show up on your world list, check to see what files were extracted. Then play!
(Source: I have a Kindle Fire HDX!)