As SLF, I probably have this entirely wrong, but let me try to put this down the way I understand it.
As far as the airframe itself is concerned, what matters is IAS, not TAS. Air density has the same effect as the wings as it does on the pitot tube, so drag, lift, stall speed, etc. will be the same at a given IAS, regardless of the density and corresponding TAS.
But when you're talking about the amount of runway needed for takeoff, what counts is the required ground speed, which corresponds to TAS (assuming calm winds). At lower density, an aircraft needs to reach a higher TAS to obtain the IAS required for takeoff. Reaching that higher TAS requires more acceleration and therefore more runway.