It's tricky and not always possible. Google Play Music stores your downloaded/cached music in an obscure directory (/storage/emulated/0/Android/data/com.google.android.music/files/music--simple, right?), but not everyone seems to be able to see the music files there. The mp3 files also have number codes for filenames rather than the actual track name. Consider it a soft way to discourage rampant music piracy.
You're probably better off downloading your music directly to your computer using either the Google Play Music website or the Music Manager program. On the website, you can download a given track or album a total of twice, while with the Music Manager program, you can download your music an unlimited number of times--the catch is that you have to download your ENTIRE library. Once you've downloaded the music, you can then copy the tracks directly to your phone, and your stock music player will see them.
The download limit may seem to be a pain, but all I do is keep the downloaded zip file somewhere safe, so that if I ever use up that download limit, I'll still have the files somewhere.
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