The key thing here is that a turbo extracts energy from the exhaust which would otherwise be lost to the environment, this means that it has the potential to improve the thermal efficiency of an engine.
At its simplest a turbocharger uses a paired turbine and compressor to pressurise the air inducted into the cylinders. Normally aspirated engines just use atmospheric pressure to 'fill' the cylinder which costs a certain amount of engine power as the descending piston needs to 'suck' air through the inlet port via ducting and filters, not a lot of power but a bit.
In addition having a higher pressure air charge in the cylinder means a greater mass of oxygen is available so a greater mass of fuel can be burned or the same mass of fuel can be burned more completely so you can get more energy per cycle or extract more energy from the fuel. This is especially the case at higher RPM where the ability to get enough air into the combustion chamber in the time available becomes a limiting factor.
This also means that you can get more power out of a smaller capacity engine block for relatively small overheads in terms of extra ancillary equipment especially for moderate boost pressures.
It is also worth noting that turbos can have a significant role to play in hybrid power trains, particularly as they can substantially increase the power to weight ratio of a given capacity engine at relatively constant RPM. It is also possible to use a turbo for electrical power input/output when connected to motor/generator here the turbo can output surplus power to the electrical storage system and similarity electrical power can be imputed to keep the turbine spinning in its optimum range to reduce or eliminate turbo lag.
a real world example of this is current generation F1 engines which are now getting close to 1000bhp peak output with a thermal efficiency of around 50% (better than any IC road car by some margin) with a capacity of 1.6 litres (from which typical consumer engines generate about 100bhp) albeit at the expense of a service life measured in tens of hours. 479