If you go into 'controls' in the settings menu in the game, you can click left or right and it will give you the choices of keyboard mouse, xbox 360 controller, or generic controller that you can see. What Don't Starve is doing is assigning it as a 'generic controller', which will just give you the 'button 1', 'dpad left', etc.
The program can be a bit confusing at first, so just watch a youtube video or two on how to set it up. Most games with controller support are setup to work the easiest with xbox 360 controller, and this program basically fools the game into thinking that's what you have. You probably need to download a program called xbox360ce(xbox 360 controller emulator).
A moderator may end up moving it, but it's no big deal I don't think. This is actually the forum about mods and mod tools.