A monarch is not, nowadays, the head of the government, but in a sense is the head of the country. Perhaps more properly stated, the cultural head/symbol of the country. The monarch is the personified representation of the country's history, culture and ideals - a heavy load to carry. Governments/political parties in power come and go, but the monarchy remains. It was there 1000 years ago and yet still here, a link to the past. It maintains continuity, constancy and unifies the fragments of society despite the chaos of the political world and everchanging society, guardian of the nation's traditions and identity. All the same can be said for Japan's emperor.
As an outsider trying to see the big picture, this is how I view, perhaps too idealistically, the monarch's function. While often criticized by the country's subjects, I suspect that most citizens would feel a loss should it be abolished.