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.

Generally boards are made by laminating together thin 2-layer boards with layers of "prepreg".

Throughole vias are drilled and plated after laminating. Blind vias may be produced by either controlled depth drilling or by drilling before laminating. Burried vias can be produced by drilling before laminating.

Here is a nice series of videos how PCBs are made step by step http://www.eurocircuits.com/making-a-pcb-pcb-manufacture-step-by-step