How does carbon dioxide extingush a fire?

Fire consists of Heat and the stuff that burns (usually oxygen from air and a combustible material). Removing any component extinguishes the fire. Carbon Dioxide CO2 is not flammable and pushes air (with it's highly reactive oxygen content) away, it is highly compressible and when released is very cold (aiding in reducing the heat. CO2 is not toxic (like the halons used in some fire fighting systems).