Both examples are redox reactions.
Let's take the general form of the first one (I'm not going to write the ions for Ba(OH)_2, I'll just leave it like this):
Ba(OH)_(2(aq)) + H_2O_(2(aq)) + 2ClO_(2(aq)) -> Ba(ClO_2)_(2(s)) + 2H_2O_((l)) + O_(2(g))
If you assign oxidation numbers to all the atoms involved in the reaction, you'll get
Ba^(+2)(O^(-2)H^(+1))_2 + H_2^(+1)O_2^(-1) + 2Cl^(+4)O_2^(-2) ->Ba^(+2)(Cl^(+3)O_2^(-2))_2 + 2H_2^(+1)O^(-2) + O_2^(0)
So, "Cl"'s oxidation number changes from +4 in "ClO"_2, to +3 in "Ba(ClO"_2)_2, while "O"'s oxidation nubmer changes from -1 in "H"_2"O"_2, to 0 in "O"_2.
"Cl"^(+4) + 1"e"^(-) -> "Cl"^(+3)
"O"^(-1) -> "O"^(0) + 1"e"^(-)
For the second reaction, you'll have
I_2^(+5)O_5^(-2) + 5C^(+2)O^(-2) -> I_2^(0) + 5C^(+4)O_2^(-2)
Iodine's oxidation number changes from +5 in "I"_2"O"_5, to 0 in "I"_2, while carbon's oxidation number changes from +2 in "CO", to +4 in "CO"_2.
"C"^(+2) -> "C"^(+4) + 2"e"^(-)
"I"^(+5) + 5"e"^(-) -> "I"^(0)
