The following is based on Van Buren and has not been confirmed by canon sources.
It's pretty simple, actually. Salt water gets pumped in from the lake to the boilers, the salt gets boiled out of the water, the steam condenses in those contraptions there, and the fresh water collects in the vat.
The Jericho water plant is the water treatment plant in the town of New Canaan. A group of squatters got it working in 2236.
The desalination plant is the largest and oldest building in town. It's loud, hot, and noisy inside when the plant is in operation during the day. Large pipes run around everywhere on the inside, and on the eastern exterior wall is a filling station where the water caravans fill up their barrels.
In 2253, the plant is currently being run by Jake Caulwell. He doesn't actually do any work within the plant, usually just keeping an eye on the convicts doing all of the actual labor.