무서워요 can also be translated as 'to be afraid of' when talking about oneself. For example 저는 무서워요 is "I'm scared" rather than "I'm scary".
I wouldn't learn it this way though. Try to imagine when someone says '무서워요' ...
They're really saying> "That thing we are looking at or talking about.... is scary."
As sonnert pointed out, 무섭다 is a descriptive verb (what we call an adjective in English) which generally translates "scary" or "frightening". 무서워하다 is the verb to be scared, frightened, or afraid.
Can this be translated to "I am afraid of tigers in the forest?" If not, why?
Yes, another example of why it would be nice to have context. Also, "in" is not the best literal translation beause of 의 which changes it to what is known as the "genitive" case in some languages. It is often used to show possession in Korean, or, as in this situation, to mean "of." A more accurate translation might be, "I'm afraid of the forest tiger. " or "I'm scared of the tigers of the forest /woods.
I had it "I am afraid of tigers in the wood", which ought to be correct, but it yielded a "wrong" again. The point is: I reckon "wood" (sg.) even more literal than "woods" (sg.; by the use of which the expression gets a bit more general).
숲 is literally translated as forest. Your use of "wood" is most likely being mistaken as the material instead of woods/forest.