It's pretty much a universal rule (though every rule has, of course, exceptions,) that little girls dream of being princesses. Certainly telling them fairy tales tends to enhance this desire.
Now, being a princess is really a political job description-it means you'll grow up to rule a country. But little girls rarely consider this part of princesshood. Although, being a female in a ruling family didn't necessarily mean you'd have any political power. Often princesses were raised as trading devices ("whoever can kill this giant will have half my kingdom and my youngest daughter in marriage," etc...) so really being a historical princess is not necessarily something to be desired.
But, there are other elements to it too. In a class system, no one moves out of their social status by hard work or luck. From this system comes the mindset that your class is deserved-elements of this belief we still have today, though not nearly as much. Therefore, being born a princess would be a symbol of your innate worth-that you alone are special and unique and born to privilege. That much appeals to everyone.
And, being a princess, you would have so much opportunity to do good for other people-to rule your people justly and secure peace and prosperity for your land.
But who am I kidding-99% of the desire to be a princess comes from the opportunity to wear pretty dresses and live in a castle and have all the pleasures of wealth at your disposal. And while I wouldn't, in actuality, want to live in one of these gorgeous castles (I'd hate to get immune to these surroundings...), doesn't everyone like to pretend, now and then?