The exhaust gases are split by the manifold into 1,2,3 and 4,5,6 which means that the exhaust pulses are balanced by the firing order of 1,5,3,6,2,4--sequential pulses to opposite halves of the manifold.  The two passages lead into two ports in the exhaust or turbine housing of the turbocharger.  These ports turn in a circle around the shaft of the turbo and also get smaller as they go around.  There are slots all the way around that lead into the center point, to the turbine fins, so all the gases at some point in their rotation around the shaft centerline go through the fins and out the back end of the turbine housing to the exhaust pipe. The temperature differential combined with gas flow spins the fins, and the greater the temperature differential, the faster the turbo shaft will spin.  The rate of spinning is not just airflow related, because boost, related to turbine speed, increases with fueling much more than with rpm.

The turbo shaft is spinning because the exhaust fins (vanes or blades) are turning from the exhaust going through them.  On the other end of the turbo shaft is another set of fins that move intake air, compressing it and pushing it out through a compressor housing.  This housing works in the reverse of the exhaust housing.  This compressed air is the "boost" that goes through the aftercooler and into the intake manifold.