Sheppard sketches out a situation in which an S Corporation owned by Trump is the general partner in his casino businesses, which went into bankruptcy in 1991 and 1992. That S Corporation likely lost a great deal of money. It also likely owed debts related to the casino businesses that it could not pay and that were canceled as part of the bankruptcies — and/or as part of Trump's subsequent out-of-bankruptcy restructurings, which continued into 1995.
Ordinarily, canceled debts would count as taxable income to the S Corporation. But if the S Corporation were insolvent (which you would have expected it to be, if its sole assets were its equity interests in Trump's bankrupt casinos), the debt cancellation would have been excluded from its taxable income.
This exclusion makes sense if you think about the personal case: Let's say you have $300,000 in debts and no assets, and you declare a Chapter 7 liquidation bankruptcy. Your debts would be discharged. It would be silly for the government to call that discharge $300,000 in income and seek around $100,000 in taxes from you, since you have no money.
In the situation described by Sheppard, the S Corporation's massive business losses would have passed through to Trump as personal losses. The offsetting debt forgiveness enjoyed by the S Corporation would not have passed through to Trump as taxable personal income because it was excluded from the S Corporation's taxable income. In total, Trump would have harvested a huge loss that exceeded his initial investment in and prior profits from the S Corporation.
"But wait!" I hope you are saying. "Wouldn't that put Trump afoul of the rule that his tax losses from the S Corporation can't exceed what he invested in it in the first place plus the prior profits?"
Yes, it would — except that, before the 2002 loophole fix, the debt forgiveness enjoyed by the S Corporation would have passed through to Trump
for the purposes of calculating the amount of profit the S Corporation had earned on his behalf, even though the same debt forgiveness did not pass through as actual taxable profit to him.
Sheppard refers to this as a "double dip" — the tax loophole would have allowed Trump to claim losses on his individual income tax return that were ultimately borne by creditors, not by him.