It has been years since my last chemistry class so some of the details may be wrong but here goes:
Wood, for example, out-gasses when it is heated. As the wood heats up, it releases flammable gasses. The fire you see is actually the gasses burning, not the wood.
Most flammable things are like this. The oxygen in the air is too diluted to have the combustion you are talking about so it takes the object you are trying to burn the need to ‘heat’ up to cause combustion.
Some chemical reactions, however, work much faster. Put some potassium in water, for example, and you have a fast chemical reaction of the potassium grabbing the oxygen’s extra electron then splitting the bond so the hydrogen atom is released which is highly flammable.