Specifically in regards to the British royal family, who don't really wield much actual policy power, it seems to be more a cultural/traditional curiosity. They're more like national mascots than anything.

In the case of actual monarchist government, I'm not a monarchist myself but in a certain sense I can understand the logic. Your leaders are trained basically from birth to be leaders, and institutional wealth and lack of elections makes them more difficult to bribe. If your monarch is a good and responsible person, the power to make sweeping improvements without slow, fiddly debates and negotiations is a big benefit.

The logic falls apart pretty quickly in reality though. Generations of isolation from the common people makes you ineffectual in the long term even if your intentions are pure. Then there's the difficulty of dethroning a monarch whose intentions aren't pure.