The ink cartridges are filled with ink and then any air is sucked out. That way when you install a new cartridge there are no air bubbles introduced but it also means the cartridge will not naturally fill the ink lines. A rubber seal called a cap top seals up with the bottom of the head and a single pump pulls a vacuum on the entire system at the same time. This vacuum sucks the air out of the lines and ink replaces it. Once the ink gets to the dampers on top of the head, they prevent the ink from flowing back to the cartridge which it will want to do naturally.
So now you have ink in the system and the dampers are holding the ink in place. When you go to print, the head fires and depletes the ink in the dampers. As ink leaves the damper, the damper collapses but the flexible plastic and spring metal resist the contraction which pulls new ink into the damper and thus, the head. The head is never truly closed like a valve. Fluid can flow through at any time. The damper is what does all the magic of keeping ink flowing and also from simply falling out of the head. The pump simply gets the process started.