I have a Vizio TV, but I've seen this feature on plenty of other brands as well. Why should I enable Game Mode on my TV when playing console video games? What benefit does it bring to my gaming experience? Should I leave it on all the time, or only when playing games?
Game mode is a setting on your TV that compensates for input lag mainly.
you can read This article for more information about it, but basically it tweaks the TV (usually by turning features off) so it can meet the requirements of the game.
Try it for yourself, watch TV with game mode on and off. you might see a few subtle differences in quality (color, sharpness, etc) and then do the same with games. The game might not look as good, but the input lag will be closer to 1:1.