Step 1: Method:
Assign oxidation numbers to each element and see which one decreases.
Step 2: Arsenic:
As changes from +3 to +5, so it loses 2 electrons per atom. That makes H$_3$AsO$_3$ the reducing agent.
Step 3: Bromine:
Bromine goes from +5 in bromate to -1 in bromide, gaining 6 electrons. This matches $3 \times 2 = 6$ electrons lost by 3 arsenic atoms, so the equation balances. The oxidising agent shifts from +5 to -1, option (B).
Final Answer:
Option (B).
\[ \boxed{\text{(B) } +5 \text{ to } -1} \]