To add fuel to the fire means

Concept — An idiom is a fixed expression whose meaning is figurative rather than the sum of its individual words. The standard way to recover that meaning is to read the physical picture the words paint and then transfer that picture to human affairs: what the pictured action does to a physical object is what the idiom does to a situation, feeling, or dispute.

Application — The picture here is fuel being poured onto a fire that is already burning. Fuel is not what starts a fire; it is what makes a flame that is already alight burn harder and spread further. Transferred to human affairs, the fire is a state of affairs already under way — a quarrel, an anger, an agitation — and the fuel is a word or act added to it. So the idiom describes saying or doing something that makes an already-running situation more intense. Standard dictionaries gloss it as making an already bad or heated situation worse, and Collins records "agitation" as an established sense of "excitement", so among the meanings offered here the matching one is "to increase the existing excitement".

Contrast — The other meanings offered belong to different English expressions:

• Conceited or arrogant behaviour is carried by expressions such as to blow one's own trumpet or to be on one's high horse.

• Extravagant spending is carried by expressions such as to spend money like water or to live beyond one's means.

• "None of these" would apply if no meaning offered matched the idiom; since the intensifying sense is offered here, it does not apply.

Cross-check — Substitute the meaning back into ordinary use: "His sarcastic remark added fuel to the fire" reads naturally as "his remark made the ongoing quarrel flare up further", and does not read as "his remark made him arrogant" or "his remark made him spend lavishly". The intensifying sense is the one that survives substitution.