I'm told that this is true, but I can't imagine why. It seems like the fact that there is less air would somehow make the engines less efficient... But that probably just shows how little I know about jet engines.
What matters for a jet engine is the pressure and temperature differentials between the exhaust gas and the ambient atmosphere. It is the expansion and high kinetic energy of the exhaust gas as it exits the engine that provides the thrust (and noise) of a jet (note this does not take into account the bypass portion of a turbofan).
The ambient pressure is atmospheric pressure, which for example at the surface is roughly 1000 hPa and at cruise might be 200 hPa or roughly a fifth of the pressure at the surface. The temperature at that altitude is also typically around -50 C.
The exhaust gas pressure and temperature is controlled by a few things:
As the outside pressure is dropping as we climb, to maintain the same pressure differential in the engine, we need less temperature and pressure in the engine, and one way to do this is to reduce airflow into the engine and fuel added to that air. The atmosphere takes care of reducing airflow (there is just less of it up at cruise, though this also depends on airspeed) and the FADEC takes care of adjusting the fuel flow. The net result is less fuel needed to produce the same pressure differential when the air outside has a lower pressure, e.g. cruise flight.
Some of the other answers/comments make reference to mass flow through the jet, and particularly the mass flow through the exhaust nozzle. I agree with that, but I didn't mention it directly because that mass flow is setup by the pressure gradient within the engine. I should also clarify that pressure at the nozzle will be at or very close to ambient atmospheric pressure and it is the pressure gradient between that ambient pressure and that in the hot section that establishes the mass flow rate out of the engine.
Lastly, to address the bypass ratio comment, see Lnafziger's comment. The turbofan engines on the EMB-145 are similar in that the bypass provides more thrust at sea level than cruise. This perhaps relates to increased fuel efficiency at cruise in that the N1 fan is doing less work and thus the N1 turbine is extracting less energy from the engine.
They work better at high altitude firstly because the air is cooler. Cool air expands more when heated than warm air. It is the expansion of the air that drives combustion engines.
The second reason is the low density of the air. Low density causes low drag and therefore the aircraft flies much faster at high altitude than on low altitude when it is given the same thrust. At this high speed, the mass flow through the engine is comparable to the mass flow at low speed in high density air (low altitude). The amount of energy needed is to heat the air to exhaust temperature is comparable between high and low altitudes. But since the aircraft at high altitude flies much faster, the amount of power generated is higher (power = thrust x speed) at altitude.
The difference with propeller aircraft is that at high speeds the propeller looses efficiency, and therefore the available power decreases with altitude.
For a non-math approach:
Let's think how a jet engine works and compare low altitude with high altitude flights. The engine takes air from the intake situated at the front. As you are climbing, the air becomes less dense (there is less air mass in a volume) so you need to go a little faster just so that the mass of the air coming in through the intake is the same in a given second. You will actually get the same mass flow of air at high altitudes as you will get at low altitudes, but you are actually travelling faster.
Then you compress that air, remembering that as you are now travelling faster higher up, the ram effect will help you out and compress some of that air for you, just by 'ramming' your engines into it at high speed. As you compress it you pass it to the combustion chamber where it burns. This combustion stage is the same for both high and low altitudes, although the fact that at higher altitudes air is colder actually helps a little bit, as we can burn more fuel without reaching dangerous temperatures, so that's nice.
After burning it up, the air is passed through a turbine then expelled out the back. Now here it gets a bit complicated: You see, it is more efficient to accelerate a lot of air (mass) a little bit(small dv), than to accelerate a little bit of air(small mass) to a very fast speed(dv). This means in turn that the faster the airplane moves, the better the jet propulsive efficiency gets. So as you climb, you go faster and the flow gets better efficiency, plus the lower air pressure behind you means there is less of a force pushing against your outflow.
So what do we have in low vs high flight:
Same amount of air intake, same amount of combustion, same amount of fuel used, better jet propulsion at higher altitudes and better speed at higher altitudes. You just get more bangs for your bucks at higher altitude.
For a mathematical approach:
I'm not even close to an expert. But here it goes.
Air like water is thick. Submarines are slower than boats. Jets are faster than boats. Automobiles are faster than boats.
Space has no friction because there is no matter. But I think jets still work in space. Of course they need oxygen. Just like Superman doesn't need ground friction to run fast, while some other super heroes do. Which is why I find it unrealistic how a super heroes that need ground friction are able to run so fast and make sharp turns without doing serious damage to floors.
So I'm guessing that being that the air is less thick up higher, that the easier it is to travel through. The jets don't depend on friction like propellers do. Superman doesn't need friction like Flash Gordon or Wonder Woman. So in space, Wonder Woman would be helpless because her propellers don't work while Superman's jets will work just fine.
Of course jet's need oxygen. So I'm not sure how all that works.
And something I didn't think of was what was mentioned in another post. Sound needs air. So yeah. Sound can increase friction.