Actually it's pretty simple. They figure out who needs the debt relief. They then go buy the debt for pennies on the dollar from one of the companies that has it. Then they forget about it. (I'd say they just delete the Excel file it's in, but there is lots of negative karma going on above this post about how wonderful Excel is)
From their website "Through sophisticated data analytics, RIP analyzes its medical debt portfolios to pinpoint those who are most in need of relief and then negotiates to purchase this debt for a significantly reduced amount – rates typically made available only to large organizations in the debt buying industry.
Once this debt is purchased – debtors are unburdened of this unpaid item."
So Oliver just gave them $14+ million in debt that he paid pennies on the dollar for and the are going to ignore it. Remember it's been written off (as a tax benefit) already, so it's only the debt people that care about it. Last guy that cared got paid by Oliver, so he's good to go.
I'd try and find some links but I'm currently on my phone.