Why does Roy Mustang's flame alchemy work? The show requires us to suspend disbelief to the point where Roy Mustang can alter densities of gases or induce some chemical reactions, but not necessarily change elements themselves. It argues the energy required to generate 'real' transmutations is much higher, so you need something like the philosophers stone.

Now, the animes claim he generates explosions by altering oxygen density profiles in the air and triggering a reaction by creating a spark. See under 'Abilities' here: http://fullmetal-alchemist-database.wikia.com/wiki/Roy_Mustang

However, while oxygen assists combustion, it is not flammable by itself. Hence, the flame needs to travel along a path of oxygen and combustible material from Mustang to his target. The show offers no explanation for where this combustible material comes from. 

Perhaps flammable gases or water vapor (that could be converted to hydrogen) in air could be utilized to create explosions. However, these gases are present at levels lower than 1% in the atmosphere, and concentrating them sufficiently to create an explosive needs volumes several orders of magnitude larger than what Mustang has access to in the show. (intuitively, the atmospheric water vapor in the volume contained by a large living room will not fill a glass of water).

Given what I have argued so far, and without having to suspend disbelief further, how does Mustang's flame alchemy even work? It feels like the authors of the show are (erroneously) assuming oxygen is combustible.
All we know about how Flame alchemy works is described in the question.  As you note the description is incomplete.

That being said, we do know that creating flames in this way is a very complicated process and is a closely guarded secret.  This secret was tattooed on the Hawkeye's back by its creator and entrusted by Hawkeye only to Mustang.  A full explanation in-universe would, therefore, not make much sense.

We might never know what the exact mechanism is and it is not necessary to understand the story.

I can come up with several head-canons to justify this physics:


The mechanism does not necessarily involve chemistry. The Crimson
Alchemist creates explosions by combining contradictory alchemical
symbols.
An unstable explosive such as tetranitromethane or nitroglycerin could be formed in gaseous state.
My preferred one is for Mustang to produce large quantities of nitric oxide (NO). There is a slow reaction where 2NO -> N2 + O2 which is exothermic but NO is considered non-flammable. This is because the reaction is slow without a catalyst because the NO molecules have to collide to react.  At very high concentrations with other chemicals, a spark, and magic (alchemy), I like to imagine Mustang can induce a suitable violent reaction which I couldn't get to working in a lab.  The actual difficulty is finding a high heat of formation molecule containing only N, O, and C stable enough to disperse harmlessly without decomposing if he can't ignite it.  Several such compounds probably exist but it isn't necessarily something that if familiar or can even has been synthesized in the real world.  I suspect NO is too stable but is a starting point. NO (nitric oxide) explosions have been observed and require a spark to get started but also require some other fuel to be present.


Further speculation would constitute a fan theory so would be off topic on this site. I hope, however, this is enough to show you that an explanation is conceptually possible if not available (or possibly even known by the creator).