Tony Foale and Vic Willoughby wrote a book about motorbike design that has probably the best insight into why a single track two-wheeled vehicle (bike) wobbles.
Firstly as the handlebars/forks etc turn (wobble) from side to side the front end of the frame will rise and fall. This can be seen when stationary if kneeling at the side of the bike facing the bike (looking at the head tube top tube) and turning the bars slightly to-and-fro. You should be able to see that the headtube toptube fall slightly 3 or 4 millimetres when the bars are turned left or right and rise as the bars are returned to the centre/staightahead position. In a speedwobble this includes the rider and frame rocking up and down slightly, pivoting about the rear hub.
That was the easy bit to explain. Now for the gyroscopic effect of the front wheel. Take a front wheel hold the ends of the axle in each hand between thumb and index finger now spin the wheel a quickly. Still holding the axle use forefinger of one hand grasp spokes and spin wheel….. easier to do than to describe. You now have a big gyroscope in your hands, try to tilt the wheel to one side the wheel tries to counteract this movement. The gyroscopic effect.
Does this seem complicated? That is because it is.
You have now observed two of the components of the speedwobble. Firstly if the handlebars start to turn away from straight-ahead position then the weight of the rider will act to push down on the frame (standing up or sitting down has no effect on gravity). Secondly the inherent stability (ironically it is the stability of the bike that causes the wobble) of the bike created at speed by the two wheels spinning as gyroscopes create a restorative force that brings the front wheel back to the straightahead position. Unfortunately this restorative force is so great (too much) that it overshoots (over compensates) and takes the handle bars too far to the opposite side. This cyclical (almost pendulum) effect then increases in intensity to produce the speedwobble.
http://www.youtube.com/watch?v=RmtoOmq3ppw
There are steering dampers available for motorbikes.