Air conditioning causes a large drop in cabin temperature, air humidity present in the air condensates into water droplets. It is similar to mist, fog and clouds. If water droplets enter in contact with a solid they form dew and moisture. Accurately, it is an aerosol of water droplets.
Water droplets appear in air when air temperature is lowered below its dew point (see further down). This phenomenon is exactly the same than when a freezer is opened and air over the freezer is in contact with cold air from the freezer tank:
Marcellus 𝚜̶𝚞̶𝚒̶𝚝̶𝚌̶𝚊̶𝚜̶𝚎̶ freezer. Source.
In your case, before air conditioning is started the cabin is filled with air from the outside, which happens to be warm and nearly saturated in water vapor. The ambient temperature is close to the dew point. The injection of colder conditioned air in the cabin decreases the temperature around the AC vents (inlet air on the picture below):
Air circulation in large pressurized aircraft, from this answer
The temperature quickly goes below the dew point causing invisible gas (water vapor) condensation into visible liquid water droplets (fog).
Due to the limited instantaneous capacity of the air conditioning system, the temperature is decreased only around the AC vents, not in the whole cabin. When the water droplets mix with the warmer air in the rest of the cabin, they immediately return to the gaseous form (vapor), and doing this they also make this air a bit colder (evaporative cooling).
As pointed out by @JanHudec the humidity level must be kept low during the flight to prevent structure oxidation. Humid air from the cabin returns through the decompression panels to the air conditioning system where it is mixed with dryer air from the packs. After some cycles the cabin air is rather dry and condensation doesn't occur anymore.
Another case of condensation:
Why does water vapor condense?
The quantity of water vapor in air is named water saturation ratio (or relative humidity) and is expressed as a percentage: 0% for dry air, 100% for air fully saturated. Saturated air cannot contain more water vapor. If vapor is added then it immediately condenses into visible water droplets.
Saturation occurs at certain combinations of temperature and pressure. For a given pressure, the higher the temperature, the larger the quantity of vapor than can be present without condensation. So condensation can occur when the temperature decreases.
Psychrometrics, the branch of physics studying how liquids and solids boil or condense, revolves around the notion of equilibrium vapor pressure, but for us mere mortals, the notion of dew point is more practical.
The quantity of vapor air can contain before being saturated is called its water vapor capacity, it increases exponentially with temperature:
Let's define two cases:
Full saturation: If the quantity of vapor is the maximum allowed for the current temperature (100% relative humidity), decreasing the temperature or adding vapor will cause some vapor to condensate into water.
Partial saturation: If air is not fully saturated, there is a (lower) temperature for which it becomes saturated. This temperature is referred to as dew point. So whatever the current temperature, fog can be created by decreasing the temperature until the dew point is reached.
For sake of accuracy: When water vapor is present in air, it is continuously condensed and evaporated. The dew point is the point where condensation occurs at a greater rate than evaporation and water droplets start accumulating.
Whether fog will form depends on the the dew point spread, that is the difference between current temperature and dew point. When the spread is large, the change in humidity or in temperature must be large. In your case, the spread is small, because air is wet and temperature is high.
Before air conditioning system is started, cabin air is the same than at the airport: 28°C and 85% humidity. AC is going to inject air at 20°:
- Quantity of vapor for 85% RH at 28°C is 23g per kg (85% * 27).
- Vapor capacity at 20°: 17g per kg.
When air is injected and temperature is decreased to 20° around the vents, air becomes fully saturated, and the 6g of vapor in excess per kg of air are condensed into fog.
Cabin air relative humidity and AC fog
This air returns to the AC system, and is mixed with air from packs which condition air entering the cabin at the 20% RH targeted for the flight. The 13.6g/kg of vapor in excess (for 20% RH) are progressively extracted by outflow valves, which regulate pressure in the cabin by releasing overboard as much air as packs push in. After some cycles, air contains only 3.4g/kg of vapor (the cycle is simplified, actually we would also account for passengers perspiration and breathing, fog re-vaporization and AC air humidity)
For typical cabin of 150m3 containing about 175 kg of air, 3.5 kg of water are extracted.
Fog in the aircraft is indeed a problem, fog outside the aircraft is another one as visibility is strongly impeded. It requires flying using Instrument Flying Rules (IFR). IFR is allowed only with a specific pilot rating and aviation certification. Knowing the dew point is important to plan a flight, in particular to know whether IFR will be required. Therefore the dew point is measured and published as aeronautical information.
Measuring the dew point
The dew point may be determined with a psychrometer (a system with dry and wet thermometers):
Whirling (how to use it) and digital psychrometers
Pressure, dry and wet temperatures, dew point and relative humidity are linked by the mean of vapor pressure. By knowing three elements, the two other can be determined. So knowing pressure, dry temperature and wet temperature allows to determine the dew point. It exists precalculated charts (Mollier diagrams) to read dew point directly. There are also online calculators.
Here is a vertical profile of the temperature and moisture information (emagram) collected by a radiosonde in Ireland:
Emagram from Valentia Observatory (Ireland). Source meteociel.fr
The vertical scale is the pressure (hPa on the left, equivalent altitude on the right) and temperature is on an oblique scale, with the quantity of water at saturation for this temperature. The dry temperature is shown in read, the wet temperature in blue, and the computed dew point in cyan. When the dew point curve is close to the temperature, moisture is hight (%RH on altitude scale) and cloud/fog can occur (like under 1,600 m for this example).
Dew point and fog/mist information
KLAX ASOS report:
As previously mentioned, when the dew point is close to the ambient temperature and the humidity percentage is high, the visibility is low and fog occurs.
Common examples of condensation
Frost around the freezer pipes: Air with water vapor enter the fridge when the door is open. When air starts to be cooled below the dew point by the pipes, condensation occurs and ice forms around them.
Air exhaled when the weather is cold. Air in our lungs is not saturated due to the high body temperature, but becomes saturated when it starts cooling.
Air saturated by water vapor coming from a pressure cooker.
As pointed out by @RyanMortensen, saturated air on car windows (when the temperature decreases during the night, or when air is saturated without temperature change, because there is more water/rain in the air that can be evaporated naturally -- Increasing car temperature above the dew point is the solution).
Fog and clouds formation: Fog because the ground is colder than the surrounding atmosphere (early in the morning), cloud because air is cooled below the dew point while it climbs (pressure and suddenness play also a role).
When water vapor changes into ice
When the dew point is below 0°C, water vapor is not condensing into water droplets, it is converted directly into ice crystals (frost).
This effect is particularly dangerous and can lead to icing in the carburetor of a piston engine. This condition can happen in areas of low pressure and low temperature, commonly created by the sudden pressure drop in the throttle venturi, due to venturi effect and the temperature drop due to fuel vaporization.
For a typical piston engine, this effect is centered on a external temperature around 10°C but extends to a large range of temperature when the engine is slowed down (descent):
Induction icing conditions
The remedy to carburetor induction icing is to heat the carburetor and raise the due point to a safe area, especially when descending.
For jet engines, induction icing (not to be confused with flying in atmosphere with ice crystals or supercooled water droplets) can cause ice to form at the engine inlet. Blocks of ice can then break, causing damage to blades, compressor stall and/or combustor flameout. It can also cause EPR probe (PT2) reading errors and subsequent FADEC fatal behavior.