The oxidation number of iodine increases from −1 (in KI) to +5 (in KIO3). The total increase in the oxidation number of iodine is 6.
The oxidation number of Mn decreases from +7 (in KMnO4) to +4 ( im MnO2). The decrease in the oxidation number of one Mn atom is 3. For 2 Mn atoms, the decrease in oxidation number is 6.
Thus, the increase in the oxidation number of iodine is balanced with a decrease in the oxidation number of Mn if the mole ratio KMnO4:KI=2:1.