There are many theories.
The straight forward answer is that they don’t. Well at least for most Irish dances, It’s mainly in the Step Dancing that the dancers use this unusual feature of Irish dancing with the stiff-arm tight at their sides
The Irish had started their dancing behind the hedge rows, stable doors and behind the bar in pubs with their upper body straight and arms tight by their sides.
Another common idea of why the straight arm dancing is the Catholic Church. The tradition says that the clergy of that time didn’t want the young people to get too carried away. So they made the dancer carry a heavy stone in both hands so that they had to keep them by their sides when dancing with the opposite sex as to prevent them from holding hands during the dances.
The most likely explanation is it was the style preferred by the dance teachers of the 18th century. The teachers of dance made their students dance with their arms in a fixed position as it looked more dignified in their eyes.

Minimal availability of deoderant in Ireland?

@elbanditoroso Because they export so much Irish Spring to the US?

More evidence.
In case you want more.

No deodorant or hairy pits.

Most likely because its distracking plus close together ones arms might hit another beside you.
Emphasis is on their footwork.

A quick google search will tell you all about it:

https://en.wikipedia.org/wiki/Irish_dance

So they can dance close together?

Another rule dictated by the Catholic Church!!! This explains a lot.

Regardless of the reason, it is a striking contrast to their leg movement. Could it have been motivated by aesthetics?