By default the Amazon Fire TV stick has navigational clicking sounds turned on. Sometime, they can get a little loud and annoying. So the best thing to do is to simply turn them off. Luckily this is possible from the settings.
To turn off clicking sounds of your Amazon Fire TV, go to Settings > Display & Sounds > Audio and then toggle Navigation Sounds on or off. Clicking sounds will stop whenever you select anything on the screen with the remote.