I simply loved NASA post on How does a turboprop engine work?
To move an airplane through the air, thrust is generated with some kind of propulsion system. Many low speed transport aircraft and small commuter aircraft use turboprop propulsion. On this page we will discuss some of the fundamentals of turboprop engines. The turboprop uses a gas turbine core to turn a propeller. As mentioned on a previous page, propeller engines develop thrust by moving a large mass of air through a small change in velocity. Propellers are very efficient and can use nearly any kind of engine to turn the prop (including humans!). In the turboprop, a gas turbine core is used.
How does a TurboProp Engine work?
See in below link the explanation and illustration by NASA
Source: NASA (The National Aeronautics and Space Administration) website