It's just optimal (not necessarily) settings to have the TV on when you're playing games. It increases the birghtness, sharpness and contrast amongst other things.
I'm sure it has a cinema mode too, and I bet everything is dimmer if turned on.
It's better if you calibrate it yourself. It will take some time though.
You won't necessarily NOTICE any differences. However, if your TV is processing your video images, it will not be displaying the images as fast if the processing were turned off. So, you will not really notice any difference when GAME MODE is OFF, but the picture you see may be a few fractions of a second behind an unprocessed image. So, it would probably be wise to turn GAME MODE ON, if those few fractions of a second mean the difference between virtual life and death.
Game mode is useful especially on some tvs that have noticeable input lag. My samsung b650 is known for that and if i dont turn on game mode it lags and is not very good for games like fps that need quick reactions. Sometimes picture tearing happens alot because image changes quickly and tv cant process it that fast. It is recommended to turn game mode on while playing.