Some Amazon Fire TV users may find themselves stuck at a screen that say ‘Connecting to Amazon’ when trying start it up. The device is unresponsive to buttons and won’t let you go any further. Restarting only takes you back to the same screen.
First, check to see if there is an outage at Amazon. If there is an outage, you may want to simply wait for the outage to be fixed.
If there is no outage, and the Fire TV is still stuck, you may want to proceed with a factory reset.
I have experienced this problem myself. The only way to get out of this is to try to send a factory reset command to the Fire TV.
A factory reset will clear all data out of the memory of the device. Your apps will be downloaded again once you re-register the device when setting it up again.
Perform these steps to get back to where you can use your Fire TV again:
When the Fire TV logo appears, press and hold the “Back” 
button and “Right” 
on the control pad for about 15 seconds, or until you see an option to factory reset.
Once the prompt appears, select “OK“, or just wait for the countdown to finish. The Fire TV will start to factory reset.
Wait for the factory reset to complete. You will be prompted to setup and register the device. Hopefully you’ll be able to go through the setup steps again and be able to enjoy your Amazon Fire TV or Firestick again without error.
If you still have problems, let me know in the comments.
Your instructions worked like a charm, thanks very much!