How does a turboprop engine work?
The following describes reverse-flow, free-turbine type gas turbine engines such as the Pratt &amp; Whitney Canada PT6 series of turbo-prop engines that I am familiar with. There are also geared turboprop engines such as the Garrett models; I am not very familiar with these.

The Beechcraft systems description for one of the PT6 models provides the following:


  The PT6 engine has a three stage axial, single stage
  centrifugal compressor, driven by a single stage reaction
  turbine. The power turbine, counter-rotating with the compressor turbine, drives the output shaft. Both the compressor turbine and the power
  turbine are located in the approximate center of the engine
  with their shafts extending in opposite directions. Being a
  reverse flow engine, the ram air supply enters the lower
  portion of the nacelle and is drawn in through the aft
  protective screens. The air is then routed into the
  compressor. After it is compressed, it is forced into the
  combustion chamber, and mixed with fuel that is
  sprayed in through nozzles mounted around the gas
  generator case. Two spark igniter plugs are used to start combustion. After
  combustion, the exhaust passes through the compressor
  turbine and two stages of power turbine and is routed through
  two exhaust ports near the front of the engine. A fuel control system schedules fuel flow to maintain the power
  set by the gas generator power lever. Propeller speed within
  the governing range remains constant at any selected
  propeller control lever position through the action of a
  propeller governor, except in the beta range where the
  maximum propeller speed is controlled by the pneumatic
  section of the propeller governor.
  
  The accessory drive at the aft end of the engine provides
  power to drive the fuel pumps, fuel control, the oil pumps, the starter-generator,
  and the tachometer transmitter.
  
  The reduction gearbox forward of the power turbine provides
  gearing for the propeller and drives the propeller tachometer
  transmitter, the propeller overspeed governor, and the
  propeller governor.


The following diagram depicts airflow through the compressor stages, the combustion chamber, the power turbines, and the exhaust section. The propeller bolts on the front of the engine at the output shaft (left side in the diagram).



The following diagram depicts the various compressor and turbine stages. Note the physical disconnect between the gas generator and power generator turbines. This is why these types of engines are called "free-turbines": the power turbine (and by extension the propellor) can spin freely from the gas generator section.