There is an easy answer to that question: I don't know it either! Strangely enough, there is not a single concluding explanation why the Irish don't use their upper body and arms, but there are a few theories that try to explain why and some even make sense.

The 'Religion-theory'

One of the possible reasons is religion. Church councils did not like dance in general. All that waving around with arms and the upper body would only enhance seduction of the opposite sex (with all it's possible side-effects) so it was decided that the upper body and arms had to be kept to the side, not being moved. As Ireland has a history of many strong catholic-religious communities, this theory is even very plausible.

The 'Lack-of-space-theory'

A different theory decides to turn things more practical: nothing more or less than space should be the cause of the now so familiar way of dancing. The small Irish homes just did not provide for waving the arms around, and the solution was to just keep them stationery during dancing. Again, this theory has truth in it. As old Dutch houses for workers are small, many old Irish houses are as well.

The 'Pub-theory'

An equivalent theory is the one that there simply was no room in the pubs for a real stage, so the dancers performed on doors layed horizontal or even danced on the beer-drums, where waving the arms around not only would heighten the risk of colliding with a fellow dancer, but also could cause them to go out of balance. From all the possible theories, I find this one to be one of the most illogical theories.

The 'British-suppresion-theory'

One more theory was brought to the world by a contestant on the American version of So You Think You Can Dance, allthough a professional dancer spoke about this theory as well. The British were to blame!
During the occupance of Ireland in the 16th and 17th century AD, the British filed a law that made dancing illegal. To pull the chain on the British, a lot of dancing started behind barndoors, where the upper half would be opened, while the bottom half remained closed, thus requiring the dancer to keep a rigid upper body.

To my opinion an even more illogical theory than the pub-theory: the hard shoe was quite probably by far not as loud as it is today, but just does not fit in the picture. Besides that, even if you don't move the upper body, it does move as the dancer jumps and lands.

This brings me to some short history and probably the most important milepost in history on Irish dance, a year that would really mark the advance of Irish dance all around the world.