I don't think you read the summary. The mechanism is pretty clear:
1. Compared with a flat tax, a progressive tax regime expands the economy because poor people spend more of their money than rich people.
2. This expansionary effect gets multiplied because this spending recirculates around and around (the "standard fiscal multiplier" in the summary).
3. Rich people accrue most of the benefits of the expansionary effect.
4. These benefits are actually larger than the initial increase in tax burden in step 1), so while everyone ends up better off, rich people end up _more_ better off than poor people. Therefore the progressive tax scheme has the net effect of boosting income inequality.
It's not a perpetual motion machine, the gain starts because rich and poor people having different consumption rates. You could only do this until you run out of tax dollars being paid by poor people.
Really, though, what the paper is proposing is saying is that rich people benefit quite a lot from expansionary policies, so much so that they can even afford to pay to get them started. I don't think this is very controversial or even counterintuitive. The thing to realize is that the paper starts with the claim that _progressive taxation makes the economy bigger_. If you accept that, it makes sense that the rich would benefit the most from it.