These insights are absolutely fundamental to understanding virtually everything about weather and climate. Until a student has his or her mind firmly around these two concepts, he or she is not ready to understand how storms work, not ready to understand why the prevailing winds blow the way they do, not ready to understand why deserts occur where they do. Because an understanding of the relationship between density and temperature of air is fundamental to so many natural processes, yet is counter-intuitive, it is a good investment of student and instructor time to construct this understanding upwards from a solid basis in the observation of real data.
Air is made of molecules, and therefore has mass.
Barometric pressure is a measure of how much mass of air, i.e. how many air molecules, exist above the point of measurement, all the way up to the top of the atmosphere. Therefore, barometric pressure decreases with elevation. Any given volume of air has density. The density of air can vary from place to place and from time to time. The difference in barometric pressure between observation sites at different elevations is a measure of the density of air in a column of air between those two elevations.