Tightrope:The game is 4:3 and your screen is 16:9, if you want the game to fill the entire screen the aspect ratio on the game would have to change and everything would look quite weird and it would also affect the scaling of the putting grid, you would not like that. Maybe one day they make wide screen versions, but currently those bars just indicates that there is nothing more to show in 4/3 format.
I never use the in game full screen button, instead I use F11 (maybe Windows have a different key) to make a proper full screen. In this mode you do not have to "exit full screen mode" to chat or whatever the in game full screen doe not allow you to do. I also find the meter slightly better with F11.
I also have a 16:9 monitor, 1366 x 768 native resolution, and I can play either way: The fullscreen icon at the lower right of the game screen gives me a "stretched" view that fills the screen, but everything is distorted (YUCK!). However, Google Chrome uses the standard "full screen" display key of most browsers (F11) which gives me an undistorted view with black bars down the sides, about 1 inch wide on my small 18 inch wide monitor. This is the mode I prefer.