Because the cabin is pressurized.
The air on a tall mountain is very low pressure, and that causes hypoxia, or a lack of oxygen. It’s not that high altitudes have less oxygen in comparison to the rest of the air, it’s that there is just less air.
So you can fix that by compressing the air to ~sea level pressure inside the cabin.
Because airplanes are pressurized to altitudes much lower than those mountain peaks. Airplanes are pressurized to somewhere between 6,000 and 8,000 ft. You absolutely do not use supplemental oxygen when climbing mountains this height. In fact, there are major cities at those altitudes where people live their entire lives. Climbers typically only use supplemental oxygen when going beyond 26,000 feet.
If the cabin became unpressurized, then you would need bottled oxygen. Thats why they have them in case of that very thing. But what happens is the higher your altitude, the less dense the atmosphere is. Gasses are compressible, so the gasses at the bottom ( sea level) of the planet become, in a way, squished more tightly together. We evolved in this denser gaseous layer, so we kinda have to stay in it because we got spoiled on the rich supply of thick air. More or less the composition of the gasses is the same at higher altitudes, but there just isnt as much of it. Have you ever hiked a mountain and then open a water bottle? Notice all the pressure escape? You captured air at a high density then moved it into a lower density environment and now the air is trying to burst free and find pressure equilibrium. You also may have noticed that after the mountain hike, your closed water bottles are partially collapsed. Due to the captured low air pressure, the higher surrounding pressure has crushed the bottle, this now is at a pressure equilibrium (or close to it disregarding the bottles natural resistance and spring) but once you open it, you may or may not find the bottle fully or partially returns to its original shape. That rebound is just from the spring of the plastic as there is no difference of pressure within the bottle versus outside the bottle.
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