the mystery是explain的宾语.when是时间状语从句.你把主句和从句的位置交换一下,你的句子就容易理解了：When the Commanding Officer admitted (said) that one half of the station did not know what the other half was doing,the mystery was explained.
-------翻译：
指挥官承认,station裏面有一半人不知道另一半人在干什麼,这时谜底就解开了/ 谜底就得到解释了.