I've seen the phenomenon quite a few times from the cockpit to some degree or another. I don't think it is directly temperature related because I've had it occur in UK (temperate), in Central America (tropical) and in winter in southern Germany (damned cold). I think what is more important is the relative humidity of the local air.
I'm fairly certain that the "stirring" action of the rotor blades is the trigger for the air to suddenly reach 100% humidity and for visible water to precipitate out of the air.
Exactly the same thing happens when a fast jet pulls hard and visible vortices appear off the wing tips. These are often seen during flying displays, I remember this very vividly from my childhood visits to Farnborough Air Show in particular.
Presumably the vortex (wing or rotor blade) produces a local low pressure and temperature which causes the air within and around to become saturated. Or perhaps the local air was already supersaturated and needed a trigger action for water particles to precipitate, which the aircraft provides. This would explain the spreading of the visible trail behind or below the aircraft in some cases.