As soon as, Immediately after, Right after
This expression is used to emphasize the fact that something occurs immediately after something else ends. The word 무섭게 is the conjugated form of the adjective 무섭다 and this expression means that the succeeding action happens in a frighteningly short time after the completion of the previous action. The form -기 무섭게 is also used. This expression only attaches to verbs.
While this expression can be replaced with -자마자 with no major change in meaning, it carries the nuance of exaggerating an even faster transition between actions compared to -자마자.