Why does adding potassium iodide increase iodine’s solubility in water?

Adding potassium iodide increases the solubility of iodine in water because the iodide ions act as Lewis bases that react with elemental iodine molecules ($I_2$).

Specifically, the iodine molecules combine with iodide ions ($I^{-}$) to form the triiodide complex ($I_3^{-}$), as shown in the reaction:
$$I_2+I^{-}\to I_3^{-}$$

This formation of polyiodides accounts for the significantly enhanced solubility of iodine in solutions containing soluble iodides, such as potassium iodide. This process is also described by the formation of the brown complex $K{I}_3$ ($KI+I_2\to K{I}_3$).1