Let $\mathbb{B}$ be the branching graph whose vertices are isomorphism classes of all irreducible $\mathbb{C}S_n$, $n\geq 0$ modules, with an edge $W\to V$ if the iso class of a $\mathbb{C}S_n$-module $W$ occurs as a composition factor of the $\mathbb{C}S_{n+1}$-module $V$. Since restriction from $S_n$ to $S_{n-1}$ of an irreducible module is multiplicity free, there is a canonical decomposition $$ \operatorname{res}_{S_0}V=\bigoplus_T V_T $$ where $V_T$ is an irreducible $\mathbb{C}S_0=\mathbb{C}$-module, and $T$ runs over paths $V_T=W_0\to W_1\to\cdots\to W_n=V$ in $\mathbb{B}$.
The text I'm reading, (Linear and Projective Representations of symmetric groups by Kleshchev, p. 9) says that $\mathbb{C}S_k\cdot V_T=W_k$. Why is the true? I don't understand what it means for $S_k$ to act on $V_T$ if $V_T$ is only a $\mathbb{C}$-vector space, and why is would give you a certain module in the path $T$ in the branching graph.