Why does SQL Server 2012 use 6x as much space for same DB?
The backup size will be around the same size as the used space in the data file if you aren't using backup compression.

The database files total 60GB on the source server, so that's what's needed on the server where you are doing a restore. While only 9GB is in use, the file size is 60GB.

You could shrink the files down on the source server and then back it up, but shrinks are bad for performance reasons. Instead, free up some space on the restore server or add more space. Disks are cheap.