To answer this question, you have to have a basic knowledge of exothermic reactions, reactions that release thermal energy (heat) into the environment. Clearly a combustion reaction is an exothermic reaction, and it behaves like any other exothermic reaction.
If you look at the image in the first reference, you can see that the two reactants in their native state have a higher amount of energy than the two future products. This difference is the amount of energy released in the reaction.
If you look at the middle of the graph, though, you see that there is an energy "hump" that must be gotten-over if the reaction will happen! This is called the "transition state" of the reaction, and it requires more energy than what the reactants can provide without anything extra. It is this amount of energy that is provided by a spark or flame.
The spark provides some energy that lets a few molecule combust. These reacting molecules release energy in the reaction, which then allows other molecules to reach the transition state energy and combust, too! The reaction then proceeds violently until combustion is complete.
Without that spark, the first set of molecules never reaches the transition state, so none of them can combust!
Of course, there are cases of spontaneous combustion--combustion without a spark or flame. However, that mechanism is actually close to the same mechanism we just described. Once the heat inside a material allows some molecules to reach that transition state, combustion can occur.