When something is “burning” the material has actually been heated enough to produce a gas. This is called [pyrolysis](https://en.wikipedia.org/wiki/Pyrolysis). The gas is what is actually involved in the production of flame, which is just a chemical reaction that produces light and heat.
As you probably know there are [4 parts to creating “fire.”](https://fire-risk-assessment-network.com/blog/fire-triangle-tetrahedron/) Oxygen, Fuel, Heat, Chemical Chain reaction.
By applying water you are cooling the source material which prevents pyrolysis. This simultaneously reduces the heat and the fuel available to continue the burning, essentially acting on two components needed to continue the burning process.
Of course their are various other factors, like if you completely submerge it in water you are also smothering it which removes the oxygen that is available. If it is in an enclosed area and you add water and allow for steam conversion in that enclosed area the steam simultaneously absorbs eat and removes the oxygen, and reduces pyrolysis.
The short answer: Water applied to fire removes heat which prevents the fuel from being usable to continue the burning process.
EDIT: Think of this explanation in terms of a piece of wood burning. There are other states of matter and other sources of fuel other than solids like wood.
The common fire needs four things to burn.
Fuel
Oxygen
Heat
And an uninterrupted chemical reaction.
You put enough of each of those into the equation, and you get fire.
Take out enough of one of them, no fire.
Water reduces heat, and displaces the oxygen. It separates the fuel and disturbs the uninterrupted chemical reaction.
https://en.wikipedia.org/wiki/Fire_triangle
Fire requires 3 friends to survive:
1. Fuel
2. Oxygen
3. Heat
Water uses the heat from fire to itself heat up(forming water vapour) thus cooling the substance taking out 1 friend(heat)
Thus extinguishing fire.
Some might argue that it also cuts off oxygen which again takes out another friend. Making fire impossible to keep burning.
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