What does the vaccum hose do and what is a vaccum petcock

The vacuum tube connects the left carburetor to draw a vacuum to open a diaphragm valve in the vacuum actuated petcock, which then allows gas to flow past the valve down to the float bowls. This way the valve only lets gas into the floatbowls when the engine is running. It is designed to be a fuel saving device, y'know, for fuel economy.