If you recall Boyle's Law from high school physics - Temperature is proportional to pressure. Aircraft wings generate an intense area of low pressure over the top of the wing (which is what essentially generates the lift). This low pressure area is magnified when flaps and slats are deployed.

On a humid day, the lowering of the pressure over the wings can basically force the air temperature at that point to lower and reach dew point temperature, essentially forming temporary clouds that are whipped around by the moving air vortices.

Once the aircraft has passed that point, the temperature generally stabilises and conforms to surrounding air temperature, which usually dissipates the temporary condensation.

The 'twirly' bits on the wingtips is basically spillover from the high pressure under the wings to the low pressure above the wings, creating that mini tornado vortex. This is also the reason that many modern aircraft have those 'winglets' on the wingtips, to try and minimise these spillover vortices which can cause problems for trailing aircraft, as well as induce extra drag on the source aircraft.