Earlier quoted context omitted.
Air pressure is strongly correlated to temperature. Cooling the air after compression reduces pumping effort during storage, reheating it increases turbine head pressure during generation.
In other words: gas compression storage suffers from temperature losses. The compressed gas is hot, the heat dissipates from storage and that energy won't be coming back. But if you extract that heat during compression you can put in a separate storage, one that is much easier to insulate than the compression tanks, and "mix it back in" during recompression, avoiding a large part of those losses.
Yes, and you can also compress air further when it's cooled (this is why there is an intercooler on turbocharged cars, as an example), this means your density per volume of storage is higher at a given pressure.
The reality though is that this is likely less efficient than many other storage mechanisms. Additionally, this may require some sort of additional energy input for reheating if the heat storage is not capable of holding long enough to fully reheat the air on release to get the highest turbine efficiency.