Why is partial pressure of oxygen lower at high altitudes?
The partial pressure of all gases will decrease at higher altitudes because the overall pressure decreases.
The partial pressure of oxygen is calculated with the following equation:
Hence, it is mathematically true that when the total pressure of gas in the atmosphere decreases, so will the partial pressure.
The reason that the total pressure decreases is that pressure is essentially a measure of the weight of 'stuff' above you; in this case atmospheric gas. As you go higher up, you are putting more of the atmosphere below you, leaving less of it above you. Therefore the weight of gas pressing down on you decreases because there is less gas.
As the amount of oxygen in the atmosphere remains constant in the atmosphere (21%), the only factor that can affect the partial pressure of oxygen is the atmospheric pressure change.