Pain in chest and back after catching self with one hand from falling while bus did sudden turn, why could have occurred?

Multiple things: Hard to tell without a physical exam, but you could have caused some muscle spasms leading to this feeling, jarred your costocondral or costovertebral joints, subluxed a rib/ribs, jammed your shoulder causing pain to entire shoulder girdle, or irritated a nerve in your neck or back with radicular signs. Again, difficult to tell, but if you are concerned and its not getting better, see your doc.