A false bottom is kind of a screen that aids in separating the sweet wort from the mash / grains. A false bottom is usually a round piece of metal with lots of little holes drilled in it. It sits at the bottom of your mash tun, and is either curved or has legs that allow it to sit an inch or so above the bottom of the mash tun.
When you sparge, the grain forms a filter bed on top of the false bottom, allowing only the wort to filter through. Thus, your spigot doesn't get clogged with grain when you run off the sweet wort.