。
Out of sheer terror, she clung to him.
Literal
She [topic-は] terrible-ness [out of excess of-のあまり] him [to-に] clung-on.
恐ろしさ is the abstract noun form of the i-adjective 恐ろしい ('terrifying, frightful') — formed by attaching the suffix ~さ to the stem. The same pattern: 美しさ (beauty), 嬉しさ (joy), 寒さ (coldness). Coupled with ~のあまり ('out of excess of'), the construction blames the action on overwhelming emotion. 抱きつく ('cling to, fling oneself onto') is the verb for the impulsive grab-and-hold action — distinct from 抱きしめる ('clutch tight'), which is a deliberate, embracing hold.