If you run up a hill enough times, you can train your quads to pull you up the hill but lungs aren't muscle so you can't train them to do anything. They're sort of like deflated balloons: they don't do anything until you force air into them.
Simply put the mechanics of breathing: When the brain says breath in, the rib cage expands and the diaphragm drops. This creates a vacuum in the thoracic cavity and the air rushes in to fill the space. (Consequently, that's why it's harder to breath at higher elevations, the surrounding air pressure is less so your lungs fill with less air when you inhale.) When your brain says exhale, the ribcage and the diaphragm return to their anatomical position and the air is pushed out of the lungs.
What it means for your running? First slow down. Second try to really be aware if your back muscles are tight when you're running. Tight back muscles might prevent your ribcage from expanding fully. And thirdly, exhale forcefully when your left foot hits the ground, which can be difficult to figure out if you're right handed; in addition to lessening the likelihood of developing a side stitch, always thinking about your left foot can help you concentrate on your breathing.