When a shutter fires, it sweeps vertically (usually). One curtain sweeps open, and then the second curtain sweeps in the same direction to close off the light. At sync speeds or slower, the whole sensor is exposed for a brief moment, so a short flash pulse can illuminate the whole frame.
But shutters are mechanical things that can only move so fast, and trying to make it move faster would require more expensive components. So instead of making the curtains move faster, to get a higher shutter speed, the curtains act work together in unison, forming a narrow slit that sweeps across the frame. The narrower the slit, the faster the effective shutter speed. What this means is that when you go above the camera's sync speed, there is no one point in time when the entire sensor (or film strip) is exposed. You can see a black band across the picture, in the link that GrinMode posted, where one of the curtains blocked the flash pulse.
Ironically, the shutter curtains still take 1/180 sec to sweep across the frame -- at high speeds like 1/8000, the curtains just have to be really close together forming a super narrow slit, so that each pixel is only exposed for 1/8000.
This is why you need the flash unit to pulse throughout the capture, so the lighting can be even as the curtains sweep across the frame.
Actually, a full-power pulse for a typical hotshoe-mounted flash can come pretty close to 1/180 sec in total duration, but the problem there is that the intensity varies during that pulse, so you will still get uneven exposure. (What this also means is that there isn't always the same kind of benefit to supporting very high native sync speeds as you would otherwise think. But that is another topic.)
The reason that the AF-200FG doesn't support HSS is probably just a marketing decision. The same reason it doesn't support Wireless operation or tilt-and-swivel. They want to keep cost down, while encouraging you to buy a more expensive unit.