PFI/PPP isn't privatisation.
It is a form of contract where a private consortium will design, build and manage a building (school or hospital). The government pays no big capital costs up front but rather pays a fixed cost over, say, 25 years.
Labour coming into power in the 90s saw that they needed a huge building programme, as the tories had run everything down. The capital cost to build everything that was needed was high, so they went down the PFI route. Hey presto, loads of shiny new buildings.
The problems with PFI contracts arrived about 10 years later. They had signed so many, that by the late 2000s the taxpayer was paying huge sums to these contracts. Other issues came up also: there are various PFI types (BOO, BOOT, etc) and folk signing a contract in, say, 1999 weren't too fussed about what would happen 25 years later. So when the PFI contracts end, some buildings become government property, but others stay with private companies meaning new negotiations, or buying a now-oldish building.
Also, there isn't much flexibility in them. A PFI school in Newcastle closed in 2010 or thereabouts due to lack of pupils, but the council must keep paying for the upkeep of an empty school. If changes to the building are needed (as will often happen) then this is hugely difficult and expensive.
In the UK we no longer use PFI but these are still very common for infrastructure projects (civil engineering and power plants) across the world.