The unbalanced redox reaction is $$\displaystyle MnO_{4}^{-} + C_{2}O_{4}^{2-} + H^{+}\rightarrow Mn^{2+} + CO_{2} + H_{2}O$$
Balance C atoms.
$$\displaystyle MnO_{4}^{-} + C_{2}O_{4}^{2-} + H^{+}\rightarrow Mn^{2+} + 2CO_{2} + H_{2}O$$
The oxidation number of Mn decreases from $$\displaystyle +7$$ to $$\displaystyle +2$$. The decrease in the oxidation number is $$\displaystyle 7-2=5$$ The oxidation number of C increases from $$\displaystyle +3$$ to $$\displaystyle +4$$. The increase in the oxidation number for 1 C atom is $$\displaystyle 4-3=1$$ The increase in the oxidation number for 2 C atom is $$\displaystyle 2 \times 1=2$$
To balance the increase in the oxidation number with decrease in the oxidation number, multiply Mn containing species with 2 and C containing species with 5.
$$\displaystyle 2MnO_{4}^{-} + 5C_{2}O_{4}^{2-} \rightarrow 2Mn^{2+} + 10CO_{2} $$
Balance O atoms by adding 8 water molecules to the products side.
$$\displaystyle 2MnO_{4}^{-} + 5C_{2}O_{4}^{2-} \rightarrow 2Mn^{2+} + 10CO_{2} + 8H_{2}O$$
Balance H atoms by adding 16 $$\displaystyle H^+$$ ions to the reactants side.
$$\displaystyle 2MnO_{4}^{-} + 5C_{2}O_{4}^{2-} + 16H^{+}\rightarrow 2Mn^{2+} + 10CO_{2} + 8H_{2}O$$
This is the balanced equation.