Why does adding water to a saturated solution increase the number of ions present in the solution?
Consider a saturated solution of $\ce{Ag2SO4}$. The following equilibrium is attained.
$$ \ce{Ag2SO4 &lt;=&gt; 2Ag+  + SO4^2-}$$

Adding water accounts to increasing the volume of the solution. Hence more amount of solute can be dissolved since solubility of a salt depends on the amount of salt dissolved per unit volume. Therefore, there is an increase in the no. of ions produced in the solution. But as Avnish Kabaj mentioned, $[\ce{Ag+}]$ remains the same. 

Also as MollyCooL said, $\pu{K_{sp}}$ doesn't change as it is a constant at a particular temperature.