Because literally the sentence means the fear is on me (and a wee bit less literally I have the fear). That’s the way you most commonly speak about your feelings and emotions in Gaelic, using nouns naming them.
Really interesting, Silmeth. I'm from Stirling and it's no uncommon to hear folk saying 'He's giein' me the boak/grue/fear' as if these feelings are physical objects which can be owned and passed around.