It pushes/removes the air, so the fire goes out. As I read, this was something that's been researched but it's not useful for anything but blowing out candles, since in a real fire whatever's burning will just reignite since the burning material is not cooled or anything. Simply, the flame is suppressed momentarily.
Another interesting possibility is that rather than pointing your sound at where the fire is now, point it at where it is about to go. What may not be able to put out an active fire may be able to suppress its spread, which would also be an enormous win in many circumstances. Even if it could just significantly retard the fire while conventional techniques are deployed in combination, it could be a potent addition to the arsenal.
To be honest, I'm sort of skeptical, especially of the "glorified speaker" approach, but as it happens there are ways to produce some very loud noises at 30-60 Hz which is very well-established off-the-shelf tech: http://epb.apogee.net/res/rehpuls.asp This used to be a standard furnace technology in American homes. Much like cars, it would be tuned to be as quiet as possible (which I've heard could still be annoyingly loud), but just as I'm sure you've all heard a car without its muffler and marveled at the difference, I'm sure something based around this technology, only meant to be as loud as possible, could achieve some serious decibalage, quite efficiently. Probably a great deal more efficiently than converting something to electricity and then converting that to sound.
I was wondering: if it starves the fuel of oxygen, it stops the fuel from burning, even if it's still hot, yes? So if it could scale from frying-pan-level to house-level, at the very least, couldn't it be used to suppress the fire, the smoke production, and the further weakening of the structure, to allow a squad to search and rescue more safely? Then when you turn off the sound, maybe the material has cooled below the reignition point, but even if it hasn't, at that point you can fight the fire traditionally.
My understanding is that it doesn't starve the fuel of oxygen, but rather starves it of activation energy. I'm fuzzy on the chemistry, but I remember it being like this:
fuel + O2 + activation energy -> CO2 + H2O + energy
The propagating sound waves move two of the input components of the reaction.
fuel: stationary
O2: moves, but is replaced by more O2
activation energy: moves and is missing from future reactions unless/until new activation energy is introduced.
Again, I took chem long ago, but I think that's what's going on.
This could reduce the amount of water required ... previously the water is used to cool down the fuel (wood) and also to cool other water from evaporating...
Could be useful if a home fire alarm can detect a small fire and start the fight at same time in places when the use of water is not advisable (libraries).
Or you could equipate an electric ashtray with this.