These will (help) prevent a headshake or tank slapper. Typically, the less rake you have on the bike, the less stable the bike is in a straight but the better it turns in. The FZ1 has a good amount of rake but riders like to change this by raising the rear, lowering the front, or sometimes both. This makes for some GREAT corners but terrible straights.
What happens is, after hitting the apex and rolling on the throttle, the front can lift. Without enough rake, when that front tire touches back down and is not hitting the pavement in the same direction you are traveling, it will start the tank slapper. This is almost totally uncontrollable and will almost always end in a crash.
The steering stabilizers are just like a shock absorber. Without them, the car hits a bump and you just bounce up and down as you travel down the road until the car settles. Well, throw in a shock and the car hits the bump, the tire goes up, comes down and settles almost immediately. This is what the stabilizer does for the bike. It is a fluid dampening mechanism that is mounted to the steering head and the frame in many different fashions. When that tank slapper should be happening, this device will prevent it and stabilize the slapper without a crash. They are adjustable so you can set them to dampen more or less. Less for touring/city driving and more for carving canyons or track days.
Hope that helps.