Thermodynamics
Gentlemen, as a mere physical chemist and at the risk of causing more confusion, may I suggest we try to stick to what is to me the fundamental process here?
The more we compress the air the more oxygen molecules we supply per unit time to the combustion chambers so we can burn more mostly hydrocarbon fuel per unit time and release more energy to do useful work per unit time.
That's the simple central point here.
Let's dig a little deeper.
The Second Law of Thermodynamics can be expressed as
ΔG = ΔH -TΔS
The combustion products are hotter than the fuel or the air reactants so the ΔS term becomes more positive as their entropy is now larger, the negative sign above then makes the TΔS term negative.
T is large and positive again making the TΔS term negative
The combustion process (yes yes ideally stoichiometrically , we'll have that discussion if you wish but not here for the moment please) is an exothermic reaction that mostly releases energy to the surroundings in the form of heat so ΔH has a negative sign on this side of the pond.
So ΔG (the free energy) is large and negative and we have lots of free energy available to do useful work - such as provide thrust.
Finally, to go back to the original question of "why compress?", we can now say "so as to release more energy per unit time from the fuel"
CW