Doesn't "possession" refer to "holding the copyright"?
If someone illegally downloads (pirates) the copyrighted content, how do they become the possessor?
In this case it means "the possession of a copy". In Germany that means that you have the file on your storage device and are able to retrieve them (put them on a screen, make a copy or send them to someone else. 'Deleted files' are also retrievable until overwritten).