History
A more compehrensive version will be coming someday. Until then, enjoy what I wrote a hundred years ago.
Irish dance is commonly (and erroneously) referred to as 'Riverdance' in ye olde post-1995 vernacular. No Irish dancer would ever use that term outside of its proper use: a show. Riverdance is a show comprised of multiple dance forms. Irish step dance is a folk dance that evolved in Ireland and spread across the world due to, among other things, the Irish diaspora and Riverdance's popularity in the 1990s. It is characterized by a lack of upper body movement, most notably arms at the sides, and dynamic rhythmic step dancing coupled with lighter softshoe.
I'll get this part out of the way first because I know it's what the people want to know most of all: no, we don't really know why Irish dance doesn't use arms in solo dancing.
In a nutshell, there are a handful of theories (and you'll see a running theme through some).
First, that the Irish dancers who visited Queen Elizabeth I's court refused to raise their arms to her.
Second, that to avoid notice of English soldiers et al, Irish folks kept their upper bodies rigid while dancing indoors (just in case someone looked in a window and was not at all disturbed by people jumping up and down).
Third, the early Dance Masters didn't want boys and girls having too much fun together such as what was exhibited in Sean Nós dancing of the time. Some accounts claim that small objects, such as stones, were held in their hands to keep them at their sides. Other records say that dancers could place one or both hands on their hips, which we do have illustrative evidence for.
Fourth, that the Catholic church didn't want any provocative dancing.
The realistic story is likely a combination more directly linked to Irish cultural genocide and persecution than anything else. Irish dance did not begin to modernize or be fully documented until the Gaelic League began its extensive preservation of Irish history and culture in the late 19th century.
Interesting to note, though, is that the group dancing, notably céilí dancing, did and does utilize arms and occasional touching of the waists and hips of other dancers. The oldest preserved Irish dance figures also used arms (as do all figures, but of course they tend to be much more rigid, with arms at right angles and rarely any closer contact), so touch wasn't forbidden. (Figures are group dances done under the umbrella of traditional Irish dance.)
Though documentation started relatively recently, Irish dance can be traced back about as far as ballet, if not a little farther. As briefly mentioned above, legend says Queen Elizabeth I had Irish dancers come to her court. What the dance looked like at that point is up for debate. Unlike ballet, it was not danced by royalty, had no regulation, and English persecution interfered with steady documentation. Before the Gaelic League, we had Dance Masters doing their best to preserve and spread their knowledge.
Irish dance evolved steadily after formal preservation began. Competition and demonstration are an integral part of the culture, meaning that it has always moved forward as dancers (and teachers) try to outdo one another. The first governing body specifically for Irish dancing (and currently the largest and oldest), An Coimisiún Le Rincí Gaelacha, was formed in 1927. In 1969, a split within the organization formed An Comhdháil na Muinteoiri le Rinci Gaelacha (the second largest governing body today).
After the creation of the World Championships in 1970, there was no stopping the momentum towards complexity and athleticism. International dancers were now exposed to each other's styles and methods of teaching. In the 1980s in Chicago, the Trinity School and its company pushed Irish dance even more towards progressive, modern change, embracing the tradition while upping the art. The culmination of Riverdance during Eurovision 1994 was inevitable.
The Irish diasapora in America influenced much of the rhythmic dancing popularized in the 20th century, most predominantly tap and clogging. They are all vernacular bedfellows.
One of the oldest traditions in Irish dance is the feis (pronounced similar to 'fesh'; pl. feiseanna) or féile (pronounced similar to 'fayluh'). The feis is a competition more for the Irish dancer than for anyone else; in the old days, and in order to be considered a feis and not a féile, it was not just about Irish dance. Livestock, music, sewing, art, singing, and baking were also featured (and still seen today depending on the location). A féile, on the other hand, was only for dance. You'll rarely see this correctly used where it should be. Feis is now the blanket term.
As an aside, Irish dance curls are often tied to the youthful sausage curls of the mid-20th century, but dancers used to change their style based on anything trendy. There are dancers with bouffants in the 1960s, and long, straight hair in the 1970s. It isn't until the 1980s that the default became curls, and in the 1990s we had our first wigs.