How does polarity of a solvent increase conductance of electrolyte?
The important parameter is the dielectric constant (relative permitivitty) $\epsilon$ of the solvent because this controls the interaction energy $E$ between two ions at a fixed distance $r$ apart, i.e. $\displaystyle E\sim \frac{1}{\epsilon r}$. If the dielectric constant is large, for water it is 78 but for benzene 2.3 you can see that the interaction energy for benzene will be far larger than for water. This means that any pair of ions formed in benzene will be attracted to one another from a far greater distance (with the same energy) than the same ions in water would be. The chance that the ions separate is given relative to the thermal energy of the solvent which is just $RT$, so if the interaction energy is far larger than $RT$ the chance of separating is small and the ion does not dissolve and vice versa. The thermal energy of the solvent tries to jostle the ions apart.

Notes. The interaction energy is  $\displaystyle E(r,\epsilon)= \frac{Z_1Z_2}{4\pi\epsilon_0}\frac{e^2}{\epsilon r}$ where $Z$ are the charges, $e$ the charge on the electron and $\epsilon_0$ the permitivitty of free space.  The thermal energy per molecule at temperature $T$ is $k_BT$ where $k_B$ is the Boltzmann constant and is $RT$ per mole. The probability of being at separation $r$ with dielectric constant $\epsilon$ is $\sim\exp(-E(r,\epsilon)/k_BT)$.