A burning fire is a chemical reaction. One of the components that is needed to react in this chemical reaction is oxygen. Therefore without oxygen, the fire will stop burning. When you throw water onto a fire, you are both cooling it and preventing it from accessing more oxygen.

To answer this question, we need to know how the fire starts burning first. Typically, fire comes from a chemical reaction between oxygen in the atmosphere and some sort of fuel including wood, gasoline and so on. Therefore, oxygen appears as a reactant in the chemical reaction of combustion. If we don't provide the reaction with oxygen, then the reaction will not start.

Therefore, oxygen helps fire to stay burning.

Fire is a chemical reaction. It requires fuel to burn, and oxygen. It's actually the same chemical reaction that keeps you alive (which is why you would die without oxygen). The oxygen combines with the fuel to turn into a new chemical - for sugars (like wood is made out of), these new chemicals are carbon dioxide gas and water vapor.

You see fire when a chemical reaction happens, called combustion. Combustion requires energy (usually heat) and two types of matter: oxygen and a fuel. Often the fuel is carbon, like the carbon in a candle wick, in a wooden log, or in gasoline. Combustion uses oxygen to burn the fuel. Without oxygen, combustion cannot occur.