My guess would be that a steering damper is a literal damper.
To very simple approximation, a damper provides an amount of resistive force proportional to the speed at which your wheel is moving left to right.
In other words, if your front wheel is wobbling back and forth then a steering damper would be useful to you. The back and forth motion will be resisted by a drag force from the damper proportional to the speed at which the wheel is moving left to right, thereby damping out that motion. If you were to look at a graph of the direction of your front wheel, you would see something like a sine wave for an oscillating wheel, and a sine wave whose amplitude was decreasing exponentially for a damped oscillating wheel.
If your front wheel is not wobbling left to right, then the damper will do absolutely nothing except make sudden left-to-right transitions harder.
Edit: It is very possible that if the damper provides a very low drag coefficient then the effects of the damper wouldn't be noticeable unless the front wheel was moving very fast from left to right.