At the cathode potassium ions are discharged:
K++e−→K
At the anode iodide ions are discharged:
I−→12I2+e−
In aqueous conditions K+ are attracted but not discharged as it takes much less energy to reduce water molecules:
H2O+e−→12H2+OH−
Hydrogen gas is evolved.
At the anode you will still get iodine produced as above if the solution is sufficiently concentrated. You may also get oxygen from the oxidation of water:
H2O→12O2+2H++2e−