Oxygen is not very soluble in water and so cannot be carried efficiently dissolved in blood plasma. Most of the oxygen is carried by haemoglobin in the red blood cells. Thus the presence of haemoglobin in red blood cells in blood increases the bloods capacity to carry oxygen. Organisms with blood containing haemoglobin are able to deliver oxygen to cells more efficiently than other organisms with blood that has no haemoglobin. The net effect is that these organisms are more effective operators in a given environment than their competitors. This is an adaptive advantage of haemoglobin. Its ability to increase the efficiency of oxygen deliverance and transportation around the body.
At high altitudes, blood is not able to absorb as much oxygen as at sea level. The human body adapts to what is effectively oxygen deprivation by initially increasing heart rate, breathing rate, then the number of red blood cells (more haemoglobin) and then increase in the density of capillaries.
Haemoglobin molecules contain four active sites where oxygen molecules can attach and so its oxygen carrying capacity as opposed to other molecules is increased 4 fold. This ability to transport large quantities of oxygen in the tissue gives mammals a considerable adaptive advantage.
Mammals respond to this environmental change n a number of ways that enables them to maintain the adaptive advantage of rapid transport of large quantities of O2 to the tissues. This response includes breathing more deeply to obtain more oxygen, measuring the number of red blood cells circulating in the blood, providing more oxygen to the cells. Red blood cells stored in the spleen will also move into general circulation to increase oxygen saturation. These changes are reversed once the body is returned to normal altitude.