Oxygen does not get rare as you go up, it is about the same % of the air composition as at sea level. What happens, however, is that the pressure and the density go down as you go up. At the altitude a jetliner cruise (35000 ft or so), the density of the air is about 30% of what exists at sea level, but (and that is the neat part) this density is also a factor in the lift, which also depends on how much air one encounters to provide the pressure differential required to fly.
The solution? Fly faster. If you go faster, you encounter more air, and thus more oxygen, per unit time. This is actually the reason why airliner fly high: so they can fly faster.
Flying faster means you get enough air per unit of time to sustain combustion at a rate that is compatible with the thrust required to keep flying at that speed. Remember that any jet engine is made of 3 parts: a compressor at the front (that compresses air to actually a much higher pressure then what we have a sea level), a combustion chamber, and a turbine that extract some of the power from the hot gases to drive the compressor and accesories; the rest of the energy is propulsive power (I simplify; high bypass turbofan have most of their thrust coming from the by-pass air that only went through the low pressure compressor stage).
As you get further up, the requirement to fly faster still to get enough oxygen goes to above the speed of sound and eventually much higher; at such altitude an aircraft has to fly so fast and use so much energy, most engines run out of power (and you start having other problems, like kinetic heatic of the airframe). All those, and the fact that compression of the air around the speed of sound creates shock wave that take energy away, that wings may not be designed for, and so on; those considerations establish the ceiling of an airplane, the maximum altitude at which it can fly.

Just remember one thing: if you have less oxygen, say only 1/3 the amount, and you do no benefit from the high speed factor, you can still burn 1/3 as much fuel. That is what airplane do as well. Sure the thrust goes down, but as long as it is enough to balance drag, there is no need to accelerate or have excess power to climb, as the airplane is already at its peak altitude and speed.