You are correct in the sense that it is not the heat that directly spins the turbine, but it is the heat that causes the exhaust gases to move. The pressure of the exhaust gas is a function of temperature and the physical properties of the gas. As the the temperature of the exhaust gas increases, the pressure increases. Pressure is a force per unit area, so the greater the heat, the greater the force applied to the turbine.
This is why a turbocharger works best when it is as close to the engine as possible. Think about a remote mounted turbo and how it will make less power than a turbo mounted directly to the manifold. The same amount of gas is flowing to the remote turbo, but the gas will have a much lower velocity because it has had time to cool and compress closer to atmospheric pressure before reaching the turbo.
