First manuafactured on a commercial basis by Klepper of Germany in the late 19th century, folding kayaks have been popular with tourists and expeditioners alike. A commmercial Klepper folding kayak was sailed across the Atlantic Ocean by Dr. Hannes Lindemann in 1956. Other explorers have used folding kayaks to explore areas from the Northwest Passage to the South Atlantic.
Hand-crafted (homebuilt) folders tend to be built in a similar manner as the commercial versions, using similar materials. A common method is to begin with a group of about 6 tubes that run the length of the kayak that will, in the finished boat, serve as gunwales, stringers, keel and deck center beam. Joined at the stem pieces fore and aft, they are separated from one another by oval, bulkhead-like "ribs".
The skin or hull is wrapped around the framework and tightened. In some boats, frame halves are inserted into the hull through a relatively small opening, and then levered apart to tension the hull. In others, the assembled frame is inserted through a long opening in the hull which is then zipped shut, laced up, velcroed or some combination of these three. Folding kayaks typically have inflatable sponsons, long tubes sewn into the hull which, when inflated, further tighten the hull around the frame and provide some extra flotation security.
An image of a homebuilt folder by Tom Yost may be seen here: 
More of Yost's kayaks may be seen here:
And an entire website devoted to folding kayaks may be found here, including an interactive English-language User Forum for folding kayak enthusiasts worldwide:
The Complete Folding Kayaker, by Ralph Diaz, Ragged Mountain Press, second edition 2003 is viewed by most as a credible source on folding kayaks.