How can I make an Automator service run a shell script and display the output?
Nice and clean solution is to display message via Notification Center.
Just execute the following terminal command in your script. 

osascript -e 'display notification "All files converted" with title "ffmpeg converter"'


'osascript' command runs any AppleScript from terminal, 'display notification' pops up nice notification. 





Also you can use 'display alert'. It has no limitation of message length and it's easy to add OK button to it. 

osascript -e 'set alertResult to display alert "Bad news :( Error converting your video file." buttons {"OK"} as warning'




Here is Apple Script commands reference. Look for 'display notification' and 'display alert' for more information.