When you finish a song you'll see an option to listen to it (playback), and, if you've got a PS Eye plugged in, also options to view snapshots and video playback. When listening to or looking at any of these, you can choose to save them to your collection - do this for 20 different pieces, and the trophy is yours.
Note this is much easier with a PS Eye camera, since you can save 10 snapshots per performance, meaning it only takes two songs to get the trophy.