Sorry to hear about your iPhone being stolen. Unfortunately, younity was not designed to retrieve music from you phone. It takes files and libraries on your computers (e.g. your laptop) and unifies/streams it to your mobile devices.
When you are looking at your files in younity, you should see a blue icon in the lower left corner of the screen; it will reveal choices for "Download" and "Share" (don't choose either yet though). Next tap that and it will put small bubbles next to all the files in the view you are in. Select each file you want to download (a small checkmark will fill the bubble), then choose "Download".
Hope that helps. Thanks.
I'm not sure I understand the problem you are encountering, can you be more detailed please? You should be able to simply select the file to access it. If it is not downloaded at that time, it will stream from the device the file is stored on. If it is downloaded, it will play as if it is a local file (which it is). Is the file(s) you are selecting not playing/opening?