Losungsschein (old spelling Loosungsschein, orthographically expressing the long vowel) represents a certificate by local German military authority about the number male subjects eligible to military service had received in a drawing (German Losverfahren; Losung [in today’s idiom,Auslosung]) that was held to decide on who had to join active service.
The German military system (ie. the military of the predecessor political entities of the German Reich) has traditionally been based on mandatory service of male subjects. The motive behind this drawing had been the perennial surplus of prospective conscripts for active service. A high number in the drawing would spare you from active service.
The Losungsschein — in particular in case you were lucky and you fell into the high number batch — also served as a military id during the service time (= period of time while you were eligible for active service, basically determined by age).