Apologize for the long-winded response, but is a very good question that leads to a lot of related info.
In this case I'm using the term to describe a small amount of cold working that refines the edge through plastic flow/deformation. The steel actually gets pushed around a bit, but in a good way. It is very similar to the mechanics behind the use of a smooth meatpacker's steel, but a lot less dramatic - also a lot less likely to lead to brittle failure if overdone, or to have any profound effect on the terminal edge angle etc.
It works best on fine grained steels with low carbide content, though I suspect it would work OK on newer PM steels with very small carbide sizes. So best results on kitchen cutlery, carbon steel, Aus8, All of the Sandvic steels I've tried it on. It has less effect on steels like D2 and even 440C though is still quite noticeable if the edge is left fairly coarse - not so much at the finer end of things. Stuff like s30v and higher also of limited value, though it does still have a mild effect. I haven't done a ton of cross testing, but I suspect similarities with steels that respond well/poorly to a smooth steeling.
It cleans up the edge quite a bit and seems to promote longer edge life than if left as is off the abrasive surface (wet/dry or a stone), though I have no empirical data to back that up. It absolutely will make the edge cut with finer character, and the micrographs show a change that would indicate just such an effect. Keep in mind that's just from a sheet of copy paper.
The first two pics are at 640x, show CS's Aus8 on a Recon at 600 grit before and after paper burnishing. The entire bevel face exhibits some effects.
These two are from a Voyager showing the effect of a smooth steeling - included in the response just so you can see what a smooth steeling actually is doing at that scale. Effects are limited to the apex and region just behind it.
While the end results are not the same, the same forces are at work in both cases, and in both cases lead to very nice boost in cutting quality. It is not easy to induce this sort of effect with good control or predictability when using a honing steel IMHO, due to it being very unforgiving of angle control or overuse. On a Washboard it is toned down some but still a very useful effect and much easier to manage without any negative results.
I've tried to reliably get this result from other improvised means, but while it is easy to duplicate the smooth steel results on something like a car window or Pyrex, the area effect created by the Washboard is far more elusive. Just seeing this effect from plain paper gives a feel for how it works with polishing compounds - the finer abrasive bites deeper without rounding the edge off, or allowing all the abrasives to line up along a single plain and polish the edge free of all irregularities. If you want a smoother presentation of the abrasive, you generally have to use additional sheets of paper and less pressure to more closely mimic the effect of a leather strop.
Here is another illustration of what I'm talking about. First pic is an edge off the fine side of an India stone:
And this is after a handful of passes on paper wrapped around the coarse side of the same stone and dosed with Flexcut compound. A moderate amount of force was applied but nothing crazy - just enough to feel the stone surface under the paper. It refined the edge like crazy, but also left it with very little bite - slight haze along cutting edge is indication of some rounding at the apex. Is also incapable of repairing any but the most superficial wear. Compare that to the second image below showing the edge after treatment with my stock compound on a WB with probably four times the applied force - virtually no edge rounding and much more pronounced abrasive effect. This is how it fixes damage or usage wear with a lot more authority. You can get a lot of good result from instinctive use once you get a feel for it. In the video I commonly use a fair amount of force, maybe a pound or two initially, then let off to maybe a half pound - still far more force than one might use on a stone or traditional strop. The plain paper is used likewise, maybe a pound or two at the bevel angle, let off to a quarter of that with the spine slightly elevated. The plain paper can refresh the edge a few times before compound is needed again - all depends on how the knife is used.
And some more illustration, a Bark River in 12c27 at 1000x using one sheet of paper and Formax green Cro 1/2 micron compound. Second pic is with 2 sheets of paper. With one more sheet of paper and less applied force, one could emulate the effect of stropping on leather with this compound.
It just gives you an idea of how it works and why, and how it plays back to the burnishing aspect. It takes a little tinkering to get a real good feel for what it can do, but is capable of a lot of customizing from a single tool. Is also not easy to reliably produce the fine microtooth that is more typical of lapping operations, or working on a soft waterstone - both methods that require a lot more technical skill than simply stropping with a little extra applied force. While no sharpening tool is truly revolutionary, it does work in an atypical manner to get strong results.
