Why does shaking a match put the fire out?
Combustion of small materials, such as a match or birthday candle, actually involve the release of volatile vapours, which themselves burn. It is not the solid material that burns. There needs to be a minimum amount of volatile material present in this combustion zone (just above the burning match) for the ignition to occur. As the combustion process continues, heat is given off, and more volatile materials are released, which in turn continues the combustion cycle. Now, if you shake a match or blow on a candle, you rapidly disperse these volatile fuels from the combustion zone, and there is no longer sufficient fuel to ignite. It is effectively removing the fuel component of the fire triangle, for just a brief moment.

Large fires can reignite because there is sufficient heat left in the fuel to further release volatile fuels, which can either self-ignite or ignite through the presence of embers.

Blowing gently on a small wood fire increase the oxygen content into the combustion zone, without dispersing the fuel. Similarly, if you experiment with a small enough wood fire, you will see that blowing on different parts of the fire will have a different outcome: blowing on the base of the fuels will increase oxygen content, and not affect volatile fuels. Blowing on the top of the fire (where the flame starts to burn) will probably put the fire out.

Will this put out a small paper fire? That will depend on the heat retained by the burning fuel. A single piece of A4 paper if shaken hard enough will extinguish. A ream of A4 paper that has burned half way down the page will be put out this way, but could easily reignite if the paper pages are pulled apart to allow oxygen into the released volatile fuels.

Generally, forest fires are accelerated by strong winds. Winds affect forest fires in a number of ways, and increase the rate of spread significantly. This is a topic for a whole other question.