At their base, they are a way of borrowing the money for capital-intensive projects, without it showing up as 'government debt'. Instead, it shows up as debt owed by financing corporations, which are basically subsidiaries of banks. But they borrow money on the market at a rate higher than the US government, and they demand a profit too, so it's more expensive. The accountants try to justify this extra expense by claiming the risk (of a budget overrun in construction, or of maintenance costs being higher than anticipated) has been handed off to the private sector. But it hasn't, really - they structure the contracts so that the parent can walk away from it if it goes wrong, and the government has to take the responsibility - because it's vital public infrastructure (a school, a bridge etc.) that can't be allowed to fail.
They are a way of hiding the necessity to borrow money to build and maintain infrastructure. They're a good deal for the banks, a lousy deal for the taxpayer, and a convenient fiction for a government that is being criticised for having too high a national debt. If people stopped worrying about the size of the debt so much, there would be no need to use them. If you want an ulterior motive for the deficit hawks, this is it - to persuade the government to pay the private sector more to fix problems the government could fix itself.