The YouTube video shows how the Raspberry Pis are assembled on an assembly line. A pick and place machine, capable of placing 25000 parts in an hour, starts the work. The PoP assembly then places the connectors, RAM and the processor in the boards — 6 boards at a time. All this is done mechanically, and finally inspected by a human being through a computer.
This examination makes sure that all the parts are fit perfectly and work as intented. After wave souldering and placing through hole components, the board is finally examined and packed by hands.
Several engineers work every day to ensure that you get the smallest and cheapest board out there at its best quality.