The reason I was at all excited about them is friction shifting. It's easier to shift with friction between the widely-spaced 6-speed cogs than it is between the narrower 7-speed—not to mention increasingly narrow 8, 9, 10, 11. Also, Uniglide doesn't have ramps, so you get better feedback on your shifts. The advantages of cassettes over freewheels (which I did know came in 6-speed!) are numerous and well-covered by Sheldon Brown here.
I was lucky enough to find a NOS 13-28 6-speed Uniglide cassette, which is photographed above. You can see how it fits on to the universal Hyperglide/Uniglide 7-speed body: the smallest cog threads on to the freehub body, and the 6-speed one leaves a little gap, just like a 6-speed freewheel would. Useful for those of you who attach fenders with eyebolts!
Below is a brochure that doesn't interest me all that much beyond the historical significance: is this the first SIS setup manual? It came attached to a 6-speed Shimano 105 downtube shifter. It's called "Quick Adjust Information":