Sometimes the front wheel can get 'light' when you are hard on the throttle or going over some heavy duty bumps. If, while the front wheel is even a little off the ground, and turns a little before setting back down on the ground it can cause 'head shake'. It's where the front of the bike violently turns back and forth in a way that usually can't be stopped by the rider themself. It can go as far as distributing that shaking through the rider via the handlebars and then shaking the rest of the bike and then you're pretty much screwed.
What a steering damper, such as Honda's HESD, does is stop that shake very quickly before it gets serious. When the damper 'detects' headshake it tightens up the steering and gets the wheel straight again. Then it loosens up and everything is dandy.
I've had my bike get bad headshake only one time(the RC-51 is usually very forgiving on it's own[unlike some other sportbikes]) and it was kind of cool and reassuring to feel my steering damper do it's thing.
If you ride hard and like the twisties, a steering damper should be one of your first purchases. Ducati, Suzuki, and Honda put it on most of their new sportbikes already.