The principle is relatively simple. The main problem is to condense water from air by spending minimal possible energy. Our home air conditioners actually do the same job- the air is cooled there below dew point and water condensate unloaded via it's pipe out to the drain. In the other words common used air conditioners spend energy for cooling and drying of air.
As far as I can understand the invention-they built device that spends energy only (or mostly) for drying of air along with collecting of water condensate and (probably) addition of salts for obtaining water close by it's quality to the common drinking or even mineral water( humans can't drink distillted mineral water).
Simplified energy balance for the process could be represented as follows:
E- is overall energy that must be spent to condense water from air
E1-is energy that must be spent for cooling of the air up to dew-point.
E2-is energy spent for cooling of water vapor(humidity) only ,up to dew-point.
What is important is that E1>>E2
Consequently most of the energy actually applied for cooling of huge amounts of the air rather then for condensation of tiny amounts of water in it.
The same problem is exists in distillation of sea water. What you actually looking for there, that is to get rid of dissolved minerals.The energy you need to apply for rendering of dissolved NaCl (and other salts of sea water) to solid state or more concentrated solution, is nothing to be compared with energy that must be applied for evaporation of water. The solution is simple. In modern plants for distillation of sea water the energy is saved by recuperation of heat of distillate. The water distillate just transfers it's heat to inlet flow of cold sea water (that circulated in condensers etc).
Almost only lost of energy are trough heat isolation of heat transfers and pipes.
In the other words if you are going to distill water in medieval way - by heating it up to boiling point and collecting of hot distillate, you should spent, say, one kwatthour of energy for 1 Lit. of distillate. But if you should use hot distillate for preheating of cold water to be distillated, you should be able to collect,say,10 Liter of distillate per 1 kwatthour of energy.
All the described above could be applied for condensation of water from air.
Using of home air conditioners for condensation of air humidity is pretty same to medieval distillation of water, but instead of loosing of energy with hot distillate you do it with cold outlet flow of air.
All you have to do is to adjust additional powerful heat transfer to a home air conditioner, where cooled up to dew-point and water free outlet flow of air will be used for precooling of inlet flow of wet air. Such modification along with changing of power supply, volume rate of air flow and other parameters and parts of air conditioner, will cause sharp raising of amounts of water condensate per kwatt of spent electricity and dropping of it's initial effectiveness as air cooler.
In ideal case the devise should condense water from air and return it dried and almost at room temperature.