How is the mass flow between 2 streamlines derived?
Short answer: Because that's how we've defined $c_1$ and $c_2$.

Long answer: Read 2.14 carefully. As you've said, we have $f(x,y)=c$ via integration. Anderson then chooses $\bar{\psi}$ as $f$ (a convention) to get $\bar{\psi}=c_1$ and $\bar{\psi}=c_2$ for the two streamlines. However, we know that $c_1$ and $c_2$ are just arbitrary constants of integration that we can set to whatever we want. Anderson just happens to know that later down the road it will be helpful to


  define the numerical value of $\bar{\psi}$ such that the difference $\Delta\bar{\psi}$ [between the two streamlines] is equal to the mass flow between the two streamlines.


Furthermore,


  $\Delta\bar{\psi}$ equaling mass flow (per unit depth) between streamlines is natural. For a steady flow, the mass flow inside a given streamtube is constant along the tube; the mass flow across any cross section of the tube is the same. Since $\Delta\bar{\psi}$ is equal to this mass flow, then $\Delta\bar{\psi}$ itself is constant for a given streamtube.