This is the simple answer. In Korea, there are different levels of speaking. To a close friend, you could drop 요 and just say 무서워.
ex. I'm scared of dogs. 저는 개가 무서워요.
Literally it means "As for me, dogs are scary".
If you are scared of swimming, 저는 수영하는 게 무서워요. (As for me, the act of swimming is scary)