Good question. Here's my guess: Since CO2 is denser than air, it would replace O2, an oxidising agent. Thus, the combustion reaction (a redox reaction) would cease.
Also, carbon dioxide is a covalent compound and has nonpolar bonds, so it would not conduct electricity in the event of an electrical fire.
Lastly, carbon dioxide would leave no harmful residues after the fire has been extinguished.
Please let me know if I am missing something; I don't think that I have fully delved into the oxidation number concept in this reply.