When OnSong is first started when running on a device with iOS 10 or above, you are automatically asked if you want to grant OnSong access to your Media Library. This question enhances your privacy. However, if you decline to allow OnSong access to your Media Library, then it cannot access the music you have stored in your Music app.
To allow OnSong to access your Media Library, go into the Settings app on your device > Find OnSong > Media & Apple Music > Turn On. Go back into OnSong and try again.