Assume that $K_{k,m}$ is not $k$-choosable with some lists of admissible colors. Let $A_1$, $A_2$, $\ldots$, $A_k$ be sets of admissible colors in the small part (that with $k$ vertices). Choose arbitrarily colors $a_i\in A_i$ for all $i=1,\ldots,k$. The large part must contain a vertex with admissible colors $a_1,\ldots,a_k$, otherwise we may color each vertex in the large part. This yields that all $a_i$'s are distinct, in other words all $A_i$'s must be disjoint. Then all sequences $(a_1,\ldots,a_n)$ are different, and the large part must contain $k^k$ distinct vertices. On the other hand, if $m=k^k$, all $A_i$'s are indeed disjoint, and the large part contain all such $k^k$ vertices, we can not color it.

Assume that $K_{k,m}$ is not $k$-choosable with some lists of admissible colors. Let $A_1$, $A_2$, $\ldots$, $A_k$ be sets of admissible colors in the small part (that with $k$ vertices). Choose arbitrarily colors $a_i\in A_i$ for all $i=1,\ldots,k$. The large part must contain a vertex with admissible colors $a_1,\ldots,a_k$, otherwise we may color each vertex in the large part. This yields that all $a_i$'s are distinct, in other words all $A_i$'s must be disjoint. Then all sequences $(a_1,\ldots,a_k)$ are different, and the large part must contain $k^k$ distinct vertices. On the other hand, if $m=k^k$, all $A_i$'s are indeed disjoint, and the large part contain all such $k^k$ vertices, we can not color it.