BTW, there is a specific DishAnywhere app for Fire tablets, you really should use that and not the one from the Android Google Play store. Just do a search of Amazon's app strore and you will find it.
I apologize you already know this, but the screen has TWO TABS: one reading somehing like "My Ques" and the other reading something like "My Transfers." You need to press the correct TAB to see either a list of your completed transfers or the list of the shows in QUE for transfers during the transfer process.
And if you don't know, you can make the transfers go much faster if you set whatever timers to automatically prepare (meaning perform the re-encoding) as soon as the show finishes recording so that when you initiate the transfer, it is only a matter of copying (or moving in the case of premium content) to the mobile device. Transfers are pretty quick, but if you attempt to transfer something that has not yet been re-encoded, then the re-code can take some time. You can also go to any recordings on your DVR and then selcet PREPARE FOR MOBILE if it has not done so becaue that option was not selected from the Timer to occur automatically.
In my case, I have invoke the PREPARE for MOBILE option on the timers for shows I know I would want to transfer so that it will save time and allow more transfers in less time. For others shows or movies I know I don't want to watch on a tiny screen, I leave the PREPARE FOR MOBILE feature on the timer UNchecked. If I change my mind and want to transfer a recording, then I just go to the recording and select the button PREPARE FOR MOBILE so that later when I do transfer it, the transfer will go much faster.
Also, the DishAnywhere app, especially on Fire, can be a bit temperamental. While it is working fine now, in the past I've had to re-boot the tablet--MORE THAN ONCE--and even just give up for the day. Then the next day the app works as it should. I have no explanation WHY it will go buggy sleepy in a bad mood and won't work right, on RARE ocassions, just that it does. I can say that the Fire DishAnywhere App most often, vast majority, works as it should. My Android tablet with the Android DA app can also get in a bad mood, but far less frequently then with the Fire. Again, I have to say that it is the RULE that the Amazon DA app on my Fire works as it should, and I still use it as my primary DA mobile device, it works that well. Just keep trying.