You have to understand that Japanese society still has very traditional elements in it. Japanese civilization goes back thousands of years, and many traditions are maintained to the present day. The emperor still has a sway for public opinion in Japan even if the royal family has no real power. The figurehead is still significant in contemporary times, just as much as the British Monarchy is in the United Kingdom.
There is emotional significance to this role, although it is not certian that Japanese people have any emotional ties to their emperor. There is definately emotionally attachment to the royal family in Great Britian, however, British people are known for there belief in tradion and pomp and circumstance, so there may be a differnce.
Many people living in a kingdom ruled over by an emperor, king or queen have a strong affection for their ruler. They see them as part of a tradition that often stretches back for centuries and people are proud of this fact. They are the constant leaders that sit outside of and above the countries' political leaders.
Much like in Great Britain, the role of the emperor in Japan is mostly for show. Akihito saying he'll abdicate the throne, though, does mean that Japan would be making a major change to their traditions, which have formed a bedrock for their entire civilization. It will bring about an emotional change for sure.