I think you both are missing the point that, as well as the compression, what is important is the oxygen content.
Air is 29% oxygen at all temperatures but - if because of expansion due to the temperature - the air has expanded to, say, twice the volume then actual amount of oxygen, per unit of volume, will be halved.
Look at climbing a mountain - the air at the top is still 29% oxygen but you will find it difficult to breath as the air is a lot thinner (less pressure - it has expanded) and so contains an actual lesser amount of oxygen per lung-full than at ground level (more pressure - it is compacted).
That makes perfect sense, yes, as air molecules expand, they occupy more space, so in other words say you can pack 1000 air molecules in a 1 cubic cm, then as it heats up, it expands, so now it may only house 500 molecules, leaving us short changed. and because there are now only 500 molecules so that 1 cubic Cm cube now weighs half as much! Hot air balloon!
Indeed in a car engine if there is less oxygen atoms, it would restrict good combustion to yield maximum power from fuel which needs oxygen to burn thoroughly, or you would just be wasting unburnt fuel, and not getting the power you intended, so cooling it down makes it denser, and packs more oxygen atoms, so you get more lean burn and get max power out.
But, wait a minute, I thought the whole idea of a compressor (turbo) was to stuff more oxygen atoms in that 1 cm cube, so instead of packing 1000 at normal air pressure ground level, we can may be force 6000 oxygen atoms, irrespective of the temperature, so i am still confused and I am going round in loops!
let us assume that in a normal aspirated engine, at a certain RPM, it draws 10,000 atoms of oxygen in a single downward stroke, along with other atoms consisting of other gasses present in the air, it compresses that on its upward stroke, which basically means there are still only 10,000 oxygen atoms give and take 1 or two that may escape past piston rings and through dodgy valve seats, and a precise measured amount of fuel is also injected (stoichiometric )and a spark ignites the mixture . A combustion takes place, which burns the fuel releasing stored energy and as well as depleting oxygen, converting it to other unwanted components such as carbon dioxide, carbon monoxide, and other harmful gasses such as nitrogen and water vapors, plus the heat, so the amount of energy released depends on the amount of fuel, as well as the amount of oxygen atoms (air) so it still leaves me questions why intercooler is needed if turbo is indeed a device that forces air into a cylinder, thus packing hundreds of thousands of oxygen atoms by force, so why use an intercooler when air entering cylinders is going to get hot any way, for the fact engine would be hot in itself, as well as further compressing the mixture would heat that already compressed air even more.