How do turbochargers improve engine performance at high altitudes / density altitudes?
Could you also address how this benefit could become a liability at low density altitudes, for example when the density altitude is negative?
An oversimplification for sure: Turbochargers essentially compress the air going into the engine (cylinders) to maintain (near) sea level pressure internally, so the external density altitude doesn't affect the performance of the engine.
The "liability" at low density altutudes is that it is possible to over-boost the engine, potentially damaging it. See the related question for more in-depth explanation and the liability of having a turbocharger at low density altitude.
Turbochargers increase available engine performance by increasing the amount of fuel/air mixture that can be burned in the cylinders. This is accomplished by compressing the air before fuel is mixed in.
The performance of a reciprocating aircraft engine is directly dependent on how much fuel can be burned and converted into heat energy, which in turn is converted into kinetic energy (motion). The amount of fuel that can be burned is dependent on how much air (actually the oxygen in the air) is available to burn with the fuel. An increase in altitude causes a decrease in air density and therefore less air available to be burned with the fuel. In a non-turbocharged (technically a "normally aspirated") aircraft engine, the reduction in available air will cause a corresponding reduction in available power. A turbocharged engine overcomes this deficiency by compressing the air brought into the engine, thereby increasing the amount of fuel that can be burned with the air, and increasing the amount of power available.
The turbocharger will rotate at a higher RPM at altitude, compressing the air such that the amount of fuel & air entering the combustion chamber is nearly the same as at sea level. From idle to full power will take longer at altitude and it may not be possible to successfully restart such an engine at high altitude until you have descended to an altitude where pressure is more favorable to start the engine.