Check out this question here: Why does a candle blow out when we blow on it? Our breath is 16% oxygen and only 4% CO2
Basically when you shake the match violently you move the fire away from its source of fuel. The flames themselves will have some momentum which you can see from the way the flame "tilts" as you move the match. When you move the match very quickly you move the source of fuel faster than the flames can readjust course.
In addition when you move the fuel source away from the flame, the fuel source will cool down rapidly and the combustion reaction will no longer be able to take place.
With respect to forest fires, winds will spread the fire because the flames are simply too much to really be blown away. that being said, for smaller fires you can still blow them out to some extent. Check out this video: https://www.youtube.com/watch?v=uyGDxglTVgA
Obviously that uses water as well, but you notice that when the jet engines turn on the fire goes right out whereas before nothing was really happening.