WinSCP for android

As you already use ES File Explorer on your Fire Tv Stick, you can use that directly to access network resources. The app supports multiple protocols, a.o. FTP, SCP and SMB – but also cloud resources such as Dropbox and SkyDrive.

^{ES File Explorer: LAN access (source: CNET; click images for larger variants)}

Once set up, you can use that to browse remote resources, and exchange files with them.