We start by identifying the oxidation number of every atom in the equation.
stackrelcolor(blue)("+1")("K")stackrelcolor(blue)("+7")("Mn")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)("+1")("H")_2stackrelcolor(blue)("+6")("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)("+1")("K")stackrelcolor(blue)("-1")("I") → stackrelcolor(blue)("+1")("K")_2stackrelcolor(blue)("+6")("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)("+2")("Mn")stackrelcolor(blue)("+6")("S")stackrelcolor(blue)("-2")("O")_4 + stackrelcolor(blue)("+1")("H")_2stackrelcolor(blue)("-2")("O")+stackrelcolor(blue)(0)("I")_2
We see that the oxidation number of "Mn" has decreased from +7 in "KMnO"_4 to +2 in "MnSO"_4.
The decrease in oxidation number shows that the "KMnO"_4 has been reduced, so "KMnO"_4 is the oxidizing agent.
Also, the oxidation number of "I" has increased from -1 in "KI" to 0 in "I"_2.
The increase in oxidation number shows that the "KI" has been oxidized, so "KI" is the reducing agent.
