1. It's a good link to the past, tradition and culture. It gives a strong feeling of cultural and historical continuity; when everything is changing, there's a rock there that's unchanging.
2. It works with a Parliamentary system, which many folks prefer. I know many Brits would rather have a Prime Minister than a President (their job isn't really the same; their powers are not the same).
3. In the UK at least, the King serves as a religious figurehead as well; the head of the Anglican Church, and many Anglicans still feel loyalty to him.
4. Pomp and pageantry.
5. Some folks just like the Monarchical system. It's up to them, really. I don't mind it. There are many ways of doing Monarchy and nearly all of them include checks and balances; some include an elected monarch, as in the old Anglo-Saxon/pre-Norman system or the way the Pope is elected. It need not always be hereditary or absolute. Absolute Monarchy is, in Europe at least, a XVIth century idea; the modern understanding of the idea of Divine Right is not really an ancient one in Europe. It would certainly be foreign to many Mediaeval rulers, even folks like Henry II would not have dreamed of going as far as Henry VIII, for example.