A sketch of their history and instructions for constructing them are most conveniently found amongst Martin Gardner's articles. The original presentation  dating from December of 1956, introduced triangle based flexagons, attributing their development to a group of graduate students at Princeton University in the late 1930's. The article was reprinted in his first puzzle collection , which is still on sale.
His first Scientific American article was followed in May of 1958 by a presentation of similar objects based on squares , duly reprinted in a second puzzle collection , also currently available. A third offering, oriented more toward ``flexatubes'' than flexagons, appears in a still later collection .
A definitive treatment of flexagons can be found in the RIAS Technical Report  of 1962 prepared by Anthony Conrad and Daniel Hartline, two students from Towson High School (near Baltimore, Maryland). Conrad's shorter summary  still covers the essential elements. Practically the only article which delves into a mathematical analysis of flexagons rather than giving a simple description of one of them is the analysis published by C. O. Oakley and R. J. Wisner in American Mathematical Monthly in 1957 . Beyond that there are a dozen or so articles scattered through mathematical and recreational publications.
Since the Technical Report was never formally published and RIAS itself has been changed, there are only a few copies in existence, two of which have been deposited in Marhematical Recreations collections1, and one of which has been optically scanned and placed on our web page2. By and large, secondary references seem to have been lost, but those items which did get published can still be found in back issues of the corresponding periodicals. There is also a summary of recent activity .