What does it mean?? "어떻게 궁금해요?"
감사합니다.

Question

Answer

"어떻게 궁금해요?"

1. 어떻게 궁금해요? => How are you curious? / In what way are you curious?
2. 얼마나 궁금해요? => How curious are you?
3. 어떤 게 궁금해요? => What [which] are you curious about?

#1 is very odd and almost never heard in real life, #2 may be heard but rare, while #3 is common.

Answer

"어떻게 궁금해요?" doesn't make sense. 
Isn't it "어떤게 궁금해요?"
"어떤게 궁금해요?' means "What do you wanna know?"

Answer

"How curious (are you)?" I would say ;)