How are multi-layer PCBs made?
In a nutshell, to make a 4-layer board, 2 double sided boards and a separator are laminated together.  You can extend this idea to 6, 8, n-layer boards.  :-)


  I'm astounded by the precision that must be required


Amazing isn't it?
A key to manufacturing multi-layer boards is the mechanical registration system which ensures layers line up with sufficient accuracy.