Oxygen is highly electronegative, which means that it attracts electrons with a high amount of force relative to most elements. The movement of electrons is what drives nearly all chemical reactions, including combustion. Oxygen helps the removal and transfer of electrons, and or molecules such as Carbon from most of our fuels which creates CO2, from the fuel to the products of the combustion reaction.
To state that oxygen supports combustion it is an understatement. Combustion is an oxidation but a fast forward one. On the other hand an explosion is a fast forward combustion. Therefore, both of them are oxidations that release thermal and luminous energies. FUEL OXYGEN AND HEAT ARE KEY REACTANTS IN THE PROCESS: COMBUSTION! Oxygen supports combustion because by definition combustion is a reaction in which fuel reacts with oxygen or another oxidant. So oxygen is a key reactant in the process.