Why must certain functions on a group take commutators to the identity? 
My question is about the identity pictured above. Can't figure out why it is true. 

The statement is poorly worded. What is $\kappa$ a function to? For instance, if $G$ is a non-commutative group and $\kappa$ is the identity function, the result is false. If $\kappa$ maps to a commutative group, however, then note that
$$\kappa(1)=\kappa(1\cdot 1)=\kappa(1)\cdot \kappa(1)$$
so $\kappa(1)$ is the identity in the codomain. Also,
$$\kappa(1)=\kappa(a\cdot a^{-1})=\kappa(a)\cdot \kappa(a^{-1})$$
so $\kappa(a^{-1})=\kappa(a)^{-1}$ in the codomain. So if the codomain is commutative,
$$\kappa(aba^{-1}b^{-1})=\kappa(a)\kappa(b)\kappa(a^{-1})\kappa(b^{-1})=\kappa(a)\kappa(b)\kappa(a)^{-1}\kappa(b)^{-1}=1.$$
