A fire needs fuel, oxygen and heat, with a self sustaining chemical chain reaction running between them. Every extinguisher removes at least one of those, and knowing which one a unit removes explains its restrictions better than memorising them does. It is also the quickest way to work out why an agent that looks capable is refused in a particular room.
Cooling
Water and water based foams take heat out of the fuel faster than the reaction returns it, until the fuel is below the temperature at which it releases vapour. This is the only mechanism that deals with a deep seated class A fire, because the heat is stored in the mass of the material and has to be removed rather than covered.
Smothering and starving
Foam seals a liquid surface so the vapour cannot leave. Wet chemical forms a crust on hot fat that does the same and holds it while the oil cools. Carbon dioxide displaces the oxygen at the seat of the fire. A fire blanket excludes air from a small contained fire. All of these separate the fuel from the air rather than cooling anything, which is why re-ignition is the risk they share.
Chemical inhibition
Dry powder interferes with the free radicals that carry the combustion reaction forward. The flame stops very quickly, often faster than any other agent, and the fuel is still hot. That combination makes powder the fastest knockdown and the most likely to be followed by a relight if nobody stays with it.
Why this predicts the label
A unit that only cools cannot help with a liquid that will keep producing vapour. A unit that only smothers cannot help with a smouldering mattress. A unit that only inhibits the flame leaves the heat behind. Every restriction printed on an extinguisher label follows from which of these it does.
Questions people ask about extinguisher work by
Do extinguishers work by removing oxygen from the room?
No. Carbon dioxide displaces oxygen locally at the fire, and in a small unventilated space that becomes a hazard to the user, which is why the instruction is to discharge and leave.
Which mechanism is safest for a person to use?
Cooling agents are the most forgiving, because they leave the fire colder than they found it. Inhibition agents give the most dramatic immediate result and the least certainty afterwards.
Does the fire triangle still hold with four elements?
The tetrahedron adds the chain reaction as a fourth face, which is what makes powder and similar agents explicable. The triangle is the teaching version and the tetrahedron is the one that predicts behaviour.