Unless USB debugging is enabled, there's not much you can do. If it is, use adb (
Mini-SDK) to pull (copy, actually) each file to the PC. You'll have to learn a little Linux (cd and ls at least) to navigate around the filesystem.
If debugging is off, you'll need to find a shop that has a working S3 screen and is willing to plug it into your phone long enough for you to copy the files to your SD card.
(If the LED layer broken, you have no light behind the screen, but in a dark room, with a flashlight, you can still see a shadowy screen if you hold the light at the right angle, as long as the LCD layer is working. And if the touch layer is working, you can press things to do something like turn .debugging on or enable USB storage.
Next time, have all your files backed up before the phone breaks or gets lost or stolen.