All to do with relative humidity. Air can hold a given proportion of water vapour according to the ambient conditions, including temperature (absolute humidity). As the temperature varies, the absolute humidity capability varies, with the result that the relative humidity goes up and down inversely with temperature (higher temperature for the same water content results in lower relative humidity and vice versa).
If the temperature drops sufficiently for the RH to get to 100% (dewpoint) then we start to see the now excess water vapour condensing out as visible water droplets (cloud).
There are various mechanisms to achieve this situation ..
(a) add some colder air (as in your aircon example)
(b) chill the air mass (eg adiabatic cooling associated with expansion .. such as in the vortex flow associated with propellers, flap tips,
nacelle chines .. etc. .. the reduced pressure inside the vortex results in a temperature drop ... increased RH ... condensation
Similar thing is at work with cloud formation.
Worse in the tropics ... I paxed a couple of legs on a Tu154 .... got a shower rather than a cloud ...