"The girl wants to fly."
Translation:여자아이가 날고 싶어해요.
Good question. It's used for 3rd person. For you and me you can use 싶다. 나는 날고 싶어요. 너는 날고 싶어요. 3rd person he, she, it, they: 싶어하다. 내 친구는 날고 싶어해요.
I think the difference in meaning is something like this: I want to fly my friend. vs. My friend wants to fly.
싶어하다 is for other people's desires, while 싶다 is for your own.
영화 보고 싶어요. I want to watch a movie.
우리 언니가 영화 보고 싶어해요. My sister wants to watch a movie.
While I was in Korea, I accidentally told a woman "우리 친구도 먹고 싶어요." Which makes me a cannibal! I should have said 우리 친구도 먹고 싶어해요." "My friend wants to eat, too."
Edit: Just want to mention that when you're directly asking someone what they want, you would, of course, still use 싶다.
뭐 하고 싶어? What do you want to do? 싶어하다 is for 3rd person