We think it is necessary to establish a statute that smoking is prohibited in the public places,and to enforce it strictivly.
establish：制定（法律,规章等）
statute：法规,法令
smoking is prohibited in the public places.公共场合禁止吸烟
that 引导的是一个同位语从句,因为that后面的句子没缺成分,所以不是定语从句,不要搞混了.
enforce：实施,执行
strictivly：严格地