Why the gradient of $C_L$-$\alpha$ graph increases with the Mach number?
This increase in lift curve slope with Mach is commonly predicted via the Prandtl-Glauert correction for 2D airfoil:

$$C_{l_\alpha}=\frac{C_{l_{\alpha_0}}}{\sqrt{1-M_\infty^2}}$$

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.

The physical reason is the magnification of pressure change due to density change (compressibility), as understood from the isentropic relation: 

$$\frac{\rho}{\rho_\infty}=\left(\frac{p}{p_\infty}\right)^{1/\gamma}$$

where $\rho$ is density and $p$ is pressure (Ref. Drela, Flight Vehicle Aerodynamics).