Breathing is one internal function that the conscious mind can control with comparative ease. The effect of controlled breathing is almost like communication with all conscious parts of one's being saying to them, "Simmer down and listen; there is something beyond the turmoil". It is communication in action that often works when words merely go in one ear and out the other, not even changing the cognitive mind. In essence, the effect is to turn all the elements of our will toward stillness and waiting.
Yes, awe arises during the extraordinary: when viewing the Grand Canyon, touching the hand of a rock star like Iggy Pop, or experiencing the sacred during meditation or prayer. More frequently, though, people report feeling awe in response to more mundane things: when seeing the leaves of a Gingko tree change from green to yellow, in beholding the night sky when camping near a river, in seeing a stranger give their food to a homeless person, in seeing their child laugh just like their brother.