the one with the money got such a fright that he …

程度状语从句such...that...句型的意义是“太……以至于……”,本句的意思是受到太大的惊吓,以至于把包掉地上了.类似的有so...that...句型,不同之处是such后面跟名词,so后面跟形容词.fright是名次,它对应的形容词是frightened（被动态,被吓到的）或frightening（主动态,吓人的）.这句话也可以用so...that...句型改写为：The one with the money got so frightened that he droped the bag.
本句中the one是主语,one可以用来指代人,with the money是介词短语作后置定语,got是谓语,that引导的从句为程度状语从句,也称结果状语从句.