How to read Phase response graph?
The frequency response of a filter (or, in general: of a linear time-invariant system) is a complex-valued function

$$H(j\omega)=M(\omega)e^{j\phi(\omega)}\tag{1}$$

where $M(\omega)$ is the magnitude response, and $\phi(\omega)$ is the phase response. The most straightforward way to interpret those two functions is to look at the way such a system responds to a sinusoidal input signal. For an input $x(t)=A\cos(\omega_0t)$, the response is given by

$$y(t)=AM(\omega_0)\cos(\omega_0t+\phi(\omega_0))\tag{1}$$

So, as you've correctly pointed out, the magnitude defines the filter's frequency dependent gain. The phase response determines the phase shift experienced by a sinusoidal input signal.

Assuming $\omega_0\neq 0$, Eq. $(1)$ can be rewritten as

$$y(t)=AM(\omega_0)\cos\left(\omega_0\left(t+\frac{\phi(\omega_0)}{\omega_0}\right)\right)=AM(\omega_0)\cos\left(\omega_0\left(t-\tau_p(\omega_0)\right)\right)\tag{2}$$

with the phase delay $\tau_p(\omega)$ defined by

$$\tau_p(\omega)=-\frac{\phi(\omega)}{\omega},\quad\omega\neq 0\tag{3}$$

From $(2)$ we see that a sinusoidal signal of frequency $\omega_0$ experiences a delay of $\tau_p(\omega_0)$. Note that the phase delay $\tau_p(\omega)$ is different from the group delay (cf. this wikipedia article).