It is difficult enough to try to understand the motivations of our close friends and family members (teenagers' music choices, for example, are perennial mysteries to adults); trying to guess the motivations of people we have never met is almost impossible.
The first thing we should note is that interest in Richard III is part of a wider movement sometimes called "medievalism", a popular movement which includes a fascination with the history, art, and culture of the middle ages, often combined with a nostalgia for (a fairly sanitized version of) pre-industrial life. Groups such as the Society for Creative Anachronism and networks of Renaissance Faires provide attendees and members a combination of historical research, costume play, role playing, and other forms of entertainment, with a continuum ranging from serious historical reconstruction through such things as fantasy role playing games (Dungeons & Dragons) or fiction (e.g. Lord of the Rings). In this way, Richard III societies are not much different from other forms of fandom, whether for popular music stars, movies, or particular periods or places (e.g. people who reenact past wars).
The Richard III society seems to include a fair number of serious local historians, people often outside academic life who enjoy delving into the history of the places in which they live. One thing many historians do is try to look behind the stories of people on the wrong side of history, and see the truth behind the legend. Since history is written by the winners, and many of our best historical sources stem from those living under or patronized by the Tudors, who replaced the Plantagenets, the problem of whether Richard III has been fairly portrayed is an interesting one.
As for the society itself, I suspect that many members first attend events or get to know members through an interest in history of the period, and become members and stay active because they find the activities of the society interesting or they enjoy the company of fellow members of the society. While I could not imagine being part of the Richard III society, as he is not a major interest of mine, I could imagine joining other groups involved in local history or archaeology, as I am curious about the past.