Deliberately reducing the amount of PO2 circulating in the breathable atmosphere around a person -- such as Kara accomplishes at sea-level with her tent -- ultimately produces a lowered rate of hemoglobin oxygenation in the arterial blood. This condition, hypoxia, can be dangerous and can, of course, impair aerobic physical exercise -- however the trick that Kara is taking advantage of is the body's ability to undergo acclimatization, in which the body's physiology and metabolism will engage in adjustments that improve the body's ability to tolerate the low-PO2 levels through different means, such as adjusting its own acidity (to change the level of interior biochemical reactivity in order to boost absorbable oxygen levels) but also -- more importantly for Kara and her endurance training -- by improving metabolism on the cellular level and blood circulation (to maximize the value of the oxygen actually obtained) and, most importantly, by...

609). Oxygen, of course, binds to the iron in hemoglobin and is then transported throughout the body to be used -- Kara's interest is its use by those muscles most heavily used in her running. As a result of this training, when Kara comes into the more PO2-rich atmosphere of New York City for her marathon, she will have trained her body to extract more from the available oxygen in the atmosphere (mostly from the hemoglobin boost but also from the metabolic, circulatory, and acidity adjustments), and thus permit her to exercise with a higher level of endurance.
References

McCardle, WD, Katch, F, and Katch, VL (2009). Exercise physiology. 7th ed. New York: LWW.

Murphy, K. (2011). Janeway's immunobiology. 8th ed. New York: Garland…