It is not necessarily the case that air temp in a low is higher than the air temp in a high (at any given true altitude). It could be either. Further study of pressure systems and cells would help but just accept it for the purpose of this question.
Give prop swinger's answer some thought. The altimeter can only sense the pressure. It does not sense density.
But density is still relevant. I have seen explanations in terms of 'columns of air' and personally I find it a useful concept. The cool dense column of air is shorter and pressure drops more rapidly with height. The warm less dense column of air is taller and pressure drops less rapidly with height. Therefore, even if the surface pressure is the same in each column, for any given true height above the surface the pressure in the cool column will be lower and so the altimeter will indicate higher. Your altimeter is calibrated to ISA conditions so it will be correct when the OAT is standard. On a cold day it will over-read in proportion to both how cold it is and how high you are and on very cold days it becomes necessary to make adjustments when flying IFR in order to maintain safety margin above obstructions.