This compressed air means more air into each cylinder and that means more fuel can be added.
The result of extra air - with extra oxygen - is a more powerful burn within the combustion process. A turbocharged engine produces more power than the same engine without a turbocharger.
The turbocharger uses the exhaust flow from the engine to spin a turbine, which in turn spins an air pump. The turbine in the turbocharger spins at speeds of up to 150,000rpm.
The advantages include the "add-on" ability of a turbocharger to an existing engine; relatively low cost; low maintenance; and light weight. The disadvantages include some "turbo lag" - where the engine hesitates before the turbines spin high enough to produce sufficient power; additional heat in the engine bay; and complexity of exhaust and intake plumbing.
Some engines use two turbochargers either of equal size or, to reduce turbo lag, different sizes.
Most turbocharged engines - diesel and petrol - use intercoolers (effectively small radiators) to cool the air before it goes to the engine. This cool air is denser, so contains more oxygen, and maximises the power produced in the combustion process.