I'll shave armpits in the summer, but won't strive to keep them clean shaven.
Couple reasons I do it:
Needs less deodorant since 3/4 of it doesn't get stuck on hairs.
Better ventilation/heat exhaust (underarms are great coolants for body as is the space between legs near crotch).
Also, if I don' at least trim them from time to time, the hair can get long and not very healthy/clean looking. In the winter, I leave them alone though.
I do a lot of cycling in summer and ability to dump excess body heat is an important factor.