Why is the equilibrium constant given in litre per mole?
$K_\mathrm c$ is defined as (for more information see here)

\begin{equation}
 K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}}
\end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation}
  \ce{2SO2 + O2 -&gt; 2SO3}
\end{equation}

you get

\begin{equation}
 K_\mathrm c = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]}
\end{equation}

where the notation $[i] = c_{i}$ was used.
This equation tells you that $K_\mathrm c$ has the unit

\begin{equation}
 \text{unit of }K_\mathrm c = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ .
\end{equation}