One way to remember Oxidation reduction problems and to know which species oxidizes and which one reduces is by following the acronym OIL RIG.
If it LOSES electrons from start to finish it oxidized hence Oxidized I LOSE (OIL), If it GAINS electrons from start to finish (meaning from reactant to product side) it reduced hence Reduced I Gain (RIG)
Iodine in this problem lost electrons going from the Ionic state with surplus electrins.-1 to 0 on the otherside.
Bromide had an oxidation number of +5 on the reactant side ( that's what it needed to be in order to balance the molecule at a -1 charge because Oxygen takes a -2 charge in redox and oxidation number assigments)
on the product side Br became an anion at Br-, it gained electrons
Oxygen went from -2 but x3 so -6 on the reactant side to -2 on the product side. losing electrons.
Hydrogen went from H+ to total at H+2 on the product side losing electrons.
Oxygen lost the most electrons while Bromide gained the most electrons
anything that oxidizes causes the other species in the opposite direction to reduce so it is indeed the reducing agent. So Oxygen is the reducing agent
Anything that reduces "RIG" or gains electrons causes the other or opposite species to oxidize so it is the oxidizing agent. Bromide gained the most electrons, it caused the other to oxidize