To remove air or gas from the system. It is important to vent the stuffing box in vertical pumps to prevent the seal faces from running dry. If you are using dual seals you will also have to vent the area between the two seals.
You cannot vent a running centrifugal pump because centrifugal force throws the liquid to the outside leaving the air in the center of the impeller.
Self priming pumps need a vent or bleed line installed between the pump discharge flange and the discharge check valve, back to the suction source.
Without this vent the pump will not be able to deliver enough pressure to open the check valve. There will be some loss through this line when the pump is operating, but the alternative is to have someone open and close the valve every time you start the pump or install some type of an automatic valve that will open and close at the correct time.