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You can't directly play Amazon movies and TV shows from the Amazon Video App on your Galaxy S6 to the Fire TV, but you can use your Galaxy S6 to remotely control the Fire TV.

On a Galaxy S6 phone like the Galaxy S6 Edge, first make sure the Fire TV and Galaxy phone are on the same WI-FI network.  Next go to Google Play to download and install the Amazon Fire TV Remote App.

Once installed, open the App and select the Fire TV you want to connect to.  Match the code that you see on the Fire TV and enter the code into the App.

  

The App will then go through a Quick Tutorial on all the features of the App.  You can use the App to swipe in different directions and navigate through the Fire TV menu system similar to Apple's remote control on the Apple TV (2015).

  

The App also supports voice search by saying what you want to search for out loud while pulling down from the top of the App. Be careful not to start pulling down from the very top of the screen since that will bring up the Galaxy's settings and notifications quick access screen.

  

If you pull up from the bottom the App, you can see additional playback controls like play/pause, fast forward and reverse.

  

After the Tutorial is done, you can start using the App.  The App also has a keyboard icon on the top right which is also convenient for typing in searches compared to using the Fire TV's physical remote.

  

The Galaxy 6 Edge works great as a remote control for the Amazon Fire TV especially if you can't find or lost your Fire TV physical remote.