How connections in a multilayered PCB used to be done?
There are no components on the inner layers of a multi-layer board. The inner layers are often used for power and ground connections, so may be nearly solid copper sheets.

Vias and holes for through-hole components are plated to provide connections between layers (from top to bottom, as well as outer layer to inner layer). 

As I understand it, a four-layer board could be made from two very thin double-sided boards, one for the top and first inner layer, the other for the second inner layer and bottom.  The two thin boards are etched as required, then glued together with a thin insulating layer between them.  The via and component holes are then drilled and plated.