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.