xanderstrike did a really good job of explaining a part of it. The monarch is our true head of state and the Royal Family our 'family of state', if you will. For all their problems, the Royal Family provides a real sense of continuity for Britishness and, to a lesser extent, an important sense of separateness from politics (usually) and what I consider critical 'reserve powers' to protect us from potential extraordinary political events (it's hard to speculate, but I doubt George V would have allowed a Hitler-esque figure to dismantle the political apparatus of the UK in the way Hitler did in Germany under von Hindenburg).

I must confess I recognize I am slightly biased as our current monarch is such a good one. I might not have been so keen on the institution in Richard III's time ;-) Politicians come and go and prime minsters are lucky to stay around for 10 years, but Queen Elizabeth II has been the figurehead of my country for 62 years and has essentially devoted her life in service of it. I have a lot of respect for her as an individual.

There are a variety of lesser, miscellaneous reasons too. I see no specific need to change, the Royal Family provides a net profit to the country financially, having an army allied to the country rather than its current political system is best, they don't have a significant impact on democratic procedure in ideal times, they provide a long-term narrative and stability, and more besides.

You know how there are Americans who hate the current government and would like to see it all thrown out, but will stand and defend their nation without hesitation? The Royal Family is a concrete and personal representation of the nation, it's like if Uncle Sam or Lady Liberty were real people who existed outside of national and local politics representing the cultural heritage of the nation as a whole. We would treat them like royalty (ha) and protect them at all costs.

Personally, I prefer a hint of the long term over a huge amount on the short term.