This short clip -- part of a longer one titled "The Hidden Complexities of a Simple Match" (http://gfm.aps.org/meetings/dfd-2014/5404ec5f69702d0771a40100
) -- uses a technique called Schlieren photography which lets you see subtle changes in the density of the air. It's used to study everything from combustion to supersonic flight.
What are you seeing here? When the breath strikes the matchhead, it pushes the column of hot gas that's emerging from the tip -- the flame itself -- to the side, so far and so fast that it can no longer convey its heat back to the match. Fire is a self-sustaining reaction: to cause any molecule of fuel to react with a molecule of oxygen, some energy is needed, and once it reacts, it emits even more energy. So long as some of that energy can be reused to ignite more fuel molecules, the reaction continues. But here, the breath prevents the heat from doing that, and so no more reactions start -- and the fire reaction is quickly extinguished.
If you watch the video at the link above, you'll see the whole story, starting from watching the match be struck, and ending with the discovery that this person failed to blow it out: enough heat persisted in the reaction, and hidden inside the matchhead, that it quickly burst back into flame.
This video won an award for the best fluid dynamics video of the year. If you want some really
cool stuff to look at, check out a gallery of all the year's winners here: http://www.wired.com/2014/11/best-fluid-dynamics-images-of-the-year/#slide-id-1659131
Via +Panah Rad