Fullscreen mode in Transformice is a method of running the game in a way that lets the player play using the full screen rather than an 800x600 windows space. Using fullscreen can be helpful in routing trolls out of hiding spaces, for showing off, or just for covering the whole screen as may be a user's preference.
To see see this option, you must have player the game for a total of at least 10 hours. Until then, you will instead see a button for the help menu.
Originally Transformice didn't support fullscreen, and unofficial Transformice clients had to be used to simulate it. Now there are three official ways to get full screen. Clicking the fullscreen button in-game while on transformice.com, using the downloadable client (found on Transformice's login screen) and maximizing window, or by opening up the game's swf directly in the window.