What exactly does κ stand for in this equation connecting solubility and molar conductivity?
Nomenclature changes and authors don't always follow the rules.  The more typical use now is $\Lambda_m$ for the molar conductivity and $\Lambda_m^0$ for the limiting molar conductivity. 

The molar conductivity, $\Lambda_m$, is given by the equation

$\Lambda_m = \dfrac{\kappa}{c}\tag{1}$

where:


$\kappa$ is the specific conductance (which is what you measure...)
$c$  is the concentration (which you know because of how the solution was made)


Now typically the specific conductance, $\kappa$, is plotted against concentration. For dilute solutions, there will be a linear relationship where the slope of the line is the limiting molar conductivity, $\Lambda_m^0$.  However at high concentrations the relationship becomes non-linear and the actual conductance is less than what the linear relationship would predict. That is because of the formation of cation and anion clusters. The local effect is thus to make the cluster uncharged (or charged less is the charges don't balance to zero). 

Now the author of your book specified a sparingly soluble salt, so at low concentrations $\Lambda_m \approx \Lambda_m^0$, and we can substitute for $\Lambda_m$ to get:

$\Lambda_m^0 = \dfrac{\kappa}{c}\tag{2}$

which can be rearranged to give:

$c = \dfrac{\kappa}{\Lambda_m^0}\tag{3}$

If the conductivity of the solution is much greater than that of pure water then this is fine. The measured specific conductance will be directly proportional to the concentration since $1/\Lambda_m^0$ is a constant. 

However the author is talking about a salt of limited solubility, so the conductivity of pure water is not negligible. Going back to the plot of specific conductance, $\kappa$, against concentration, $c$, we can determine that at 0 concentration there is some conductance which I'll call $\kappa_w$ which is the conductance of pure water. So for a salt with limited solubility the equation is:

$c = \dfrac{\kappa - \kappa_w}{\Lambda_m^0}\tag{4}$

If you look back at equation (2) you can see why equation (3) doesn't work for solutions of very low conductivity. When $c=0$ equation (2) blows up since you can't divide by $0$. 

So the author used $\kappa$ in two different ways which is very poor writing. We could define $\kappa'= \kappa - \kappa_w\tag{5}$ then $c = \dfrac{\kappa'}{\Lambda_m^0}\tag{6}$