How does ignited gas spin the turbine?
A turbine is a machine that extracts energy from a moving fluid and converts it into work. In the turbine blades, the air is expanded, which produces work. In many ways, the turbines are the opposite of compressors.



"Two stage reaction turbine" by S.M.Yahya - Turbomachinery. Licensed under Public Domain via Commons.

The turbines used in the aircrafts are axial flow turbines, which employ multiple stages to extract work from the air coming out of combustion chamber.

The turbines have two parts- the stationary nozzles (or stators) and  the moving rotors. The purpose of the nozzles is to turn the incoming flow so that that it impinges on the rotors at the correct angle, while the actual work is extracted out from fluid by the rotors.



"Velocity triangle for an axial turbine stage" by S.M.Yahya - Turbomachinery. Licensed under Public Domain via Commons.

Basically, the rotors are 'turned' by the incoming fluid as shown above which produces work to rotate the shaft (which runs the compressor). In case of turbojet engines (and to some extent turbofan), the air pressure decreases, while the velocity increases. This provides the required thrust.

The main mechanism by which the turbine blades spin is that the flow turns the rotor blades by hitting them, rather than any lift being produced.