Maximum number of configurations of Turing machine after $n$ moves
Note that you are bounding the maximal number of configurations, that the machine can be in, from above. To see it more clearly, when a TM $M$ runs on a finite word $w$ it induces a configuration tree $G = \langle V, E\rangle$ where:  


$V$ is the set of configurations that $M$ can be in when it runs on $w$,
$q_0w$ is the initial configuration of $M$ on $w$ and it is the root of the configuration tree,
and the edges in $E$ are defined such that $u$ is a child of $v$ whenever $u$ a consecutive configuration of $v$ (by consecutive we mean that it is possible to reach $u$ from $v$ without violating the transition function of $M$).


As you have mentioned, $k$ is a branching factor, and we usually mean by that the maximal branching degree in the configuration tree. In this aspect you're not correct, $k$ is at most $|{Q\times\Gamma\times \{L, R\}}| = {2|Q|\cdot |\Gamma|}$, and this follows from the definition of the transition function. Indeed, $k$ bounds from above the number of consecutive configurations (which is a subset of configurations) and not the number of all possible subsets of configurations of the TM.

Considering the configuration tree, it is a tree of height ($n = |w|$) with branching degree $k \leq {2|Q|\cdot |\Gamma|}$. Now its not hard to see that such a tree has at most $k^n$ leaves which bounds from above the number of configurations that $M$ can be in when it runs on a word $w$ of length $n$.

A thing worth mentioning is that, $k$ is a constant that does not depend on the input word $w$ (on $n = |w|$) or on a specific configuration tree, and we know that it exists. This sometimes makes life a bit easier as some algorithms/proofs rely on its existence to conclude some upper bounds.