The way they take 1 dollar to eliminate 20 in debt is this: they pretend to be a scumbag bill collector and buy the debt from someone who has given up hope of collecting for about 5 cents per dollar. They then write off the debt as forgiven.
They aren't really pretending, anyone can buy debt packages.
I would note that if they are doing this 100% in compliance with the law it can cause some minor inconvenience to the person getting their debt payed off because the IRS considers forgiven debt to be income, potentially raising the persons tax burden the next year.
As listed in the FAQ, Rolling Jubilee is a non-profit 501c4 and thus does not file a 1099-C with the IRS related to debt cancellation. (Or, in English, they're a non-profit social welfare organization, not a charity, so the IRS does not consider the forgiven debt income.)
Actually, under IRS Code 64(a)(4) it is always income if debt is forgiven. You are rightly pointing to the practical difficulty of tracking that in the absence of a 1099C form. However, in any event IRS Code 108(d) deals with insolvency provisions for debt forgiveness. Also, this is a nasty surprise of some bankruptcies as I understand it.
Intellectually, loans aren't income because though you get something, it has to be paid back. Remove the payback and it's income....
Practically, the reason they enacted this rule was because rich people were abusing the daylights out of the situation. They would "loan" their kids large sums of money, sometimes in the millions of dollars range, and then simply "forgive" it. Thus, they would avoid paying any taxes, estate, capital gains, etc.... Now, if the IRS could be persuaded to grant an exception limited in the number of times and dollar amount of forgiveness, or limiting the source of such forgiveness to non profits and disinterested parties, that might work. Who knows.