From the article you cited:
A fan draws in humid air and carries it through a refrigerated evaporator.
Evaporator is just a heat exchanger in which the working fluid (refrigerant, inside the tubes) evaporates. It accomplishes this by sucking out the heat from the air flowing through. This is the rectangular portion on the right side of the apparatus in the image.
The air is cooled well below its dew point.
The dew point is the temperature of the air at which it can't carry any more water vapour than it already does. If you cool this air, the excess water vapour
The water condenses on the cold surface of the evaporator and drips into a water container or is led directly to a drain.
This is pretty self-explanatory. You now have cold air, minus the water vapour.
Then the cold dry air continues through a hot condenser which heats it up and returns it to the room to pick up new humidity.
The cold, dry air from the previous step can't be introduced into the room as-is (this would mean the apparatus is a cooler). Remember the working fluid that was in the pipes of the evaporator, which sucked out heat from the incoming air? That fluid is pumped to another heat exchanger called the "condenser". Here, this fluid (which is a gas currently), is condensed into liquid. In doing so, it throws out its excess heat, which is taken by the "cool, dry air" flowing through the condenser, rising to slightly above the original temperature. This is for two reasons, because some work is involved in running the system and because the water gives off some latent heat in order to change phase. This air is then sent to the room.
This procedure is continued until the desired condition is achieved.
The end result is (slightly-more-than-)room-temperature air, minus the moisture.