When solid $KI$ is placed in water, partially negative oxygen atoms (from the water molecules) attract positive $K^+$ ions. At the same type, the partially positive hydrogen atoms attract negative $I^-$ ions. This process takes the ions from the solid to the solution. After that, these ions undergo hydration as water molecules surround $Li^+$ and $Br^-$, keeping them in solution.
Work Step by Step
You can see the necessary information on the section "Solutions with Ionic and Polar Solutes", page 328.