The oil is all about corona suppression, it probably helps to cool the coil too, through convective cooling. But corona suppression is the big thing and the corona must be suppressed in any high-voltage coil or transformer.
Corona is a blue glow that forms around conductors at high electrical field intensity. Corona is the electrical breakdown of air. In corona the air ionizes and forms all sorts of nasty, electrically charged molecules. These nasty molecules can attack and destroy the insulation in the coil. Then the coil burns out.
If you exclude the air, the corona can't form. One old-school method is to use an oil filled coil. Newer coils are epoxy impregnated. It does the same job, excludes air and suppresses corona.