PFI was brought in by the Major government. At the time, there was an obsession with the Public Sector Borrowing Requirement. There was a taboo against it going above such-and-such a percent, though like many taboos that government obsess over, it was mainly bollocks.
PFI was a way of getting around this as expenditure did not count against PSBR. So you could have your new hospital/school/police station/whatever. It also involved the highly efficient private sector in the story, which was seen as a Good Thing.
I remember agreeing with my mate, at the time, that this was like buying a car on a credit card. That it would be ludicrously expensive. That it was obviously more sensible for the government just to borrow the money, given that it can borrow more cheaply than anyone else.
(Sigh). But the obsession with the PSBR plus the desire to have new shiny things (which was justified after years of neglect) meant that this PFI nonsense went ahead.
You will notice no one mentions the PSBR anymore. Fuck knows what it is now. But it's all of a piece with similar obsessions with the balance of payments, M3 money supply, and the value of the £ against 'a basket of foreign currencies'. All of which used to be on the TV every night, none of which have been mentioned in years. (I wouldn't even know where to look for the M3 figure these days, but they used to talk about it as if civilisation itself hung on its level.)