The fire was caused by rubbish beneath wooden escalators being ignited, thought to be caused by a dropped cigarette.
The fire was made worse by the decision to stop tube trains from stopping at the station, in an attempt to prevent people getting out from the trains into the burning station. This meant of course that passengers on the station could not escape onto trains. Instead of stopping at the platforms, the trains continued through, acting like pistons in the confines of the tunnel and forcing an air draft up the chimney-like escalator shaft, fanning the fire further. (This "piston effect" is well known, but is usually beneficial because it keeps air circulating between the deep tube stations.)
The Kings Cross fire led to the banning of smoking throughout the London Underground network, and the eventual removal of all wooden escalators from Underground stations. The publication of the Fennell Report into the fire resulted in the introduction of the Fire Precautions (Sub-surface Railway Stations) Regulations 1989.
See also Kings Cross station