As the engine runs, oil gets thrown about by the crank, cam chain, rocker arms, valves. This thrown oil creates a fog like mist.
The baffle in the breather stops most of the mist going out of the engine. Some gets out anyway. The tube lets this drain out to the ground and not on the bike.
Some early years had the hose run into a hole on the left engine cover so this oil could drain out on the chain.
Leo