Why the gradient of $C_L$-$\alpha$ graph increases with the Mach number?

From the JAA ATPL Book 13 (Principles of flight) the gradient $$\frac{dC_L}{d\alpha}$$ of the $$C_L$$-$$\alpha$$ graph increases for higher Mach numbers. Can someone explain why this is the case?

$$C_{l_\alpha}=\frac{C_{l_{\alpha_0}}}{\sqrt{1-M_\infty^2}}$$
where $$C_{l_\alpha}$$ is the 2D lift-curve slope, $$C_{l_{\alpha_0}}$$ is the 2D lift-curve slope in incompressible flow, and $$M_\infty$$ is the free-stream Mach number (Ref. Anderson, Fundamentals of Aerodynamics), and is accurate before local shock appears.
$$\frac{\rho}{\rho_\infty}=\left(\frac{p}{p_\infty}\right)^{1/\gamma}$$
where $$\rho$$ is density and $$p$$ is pressure (Ref. Drela, Flight Vehicle Aerodynamics).