Let $\lambda$ be a partition of $n$. I'm trying to show that $Res^{S_n}_{S_{n-1}}V_\lambda\cong \oplus _{\mu:\mu \vdash\lambda}V_\mu$, ($\mu \vdash\lambda$ means that $\mu$ is a partition of $n-1$, such that $\lambda_i=\mu_i$ for all $i$ except one, in which $\lambda_i=\mu_i+1$).
As a first step, let $C$ be a conjugacy class of $S_{n-1}$, given by a cycle structure, $C=(i_1,\dots i_n)$ ,s.t. $\Sigma i_k =n$, and let $C'=(i_1+1,\dots i_n)$ be a cycle structure of $S_n$. I managed to show that $\chi_{Res^{S_n}_{S_{n-1}}V}(C)=\chi_V(C')$, for every finite representation $V$ of $S_n$. It is quite clear that the next step should by applying Frobenius character formula on the right side, but I couldn't make it work.
Using Frobenius reciprocity, $$ \mathrm{Hom}_{S_{n-1}}(V_\mu,Res^n_{n-1}V_\lambda)\cong\mathrm{Hom}_{S_n}(Ind_{n-1}^nV_\mu,V_\lambda). $$ Therefore, the result you are after is equivalent to proving $$ Ind_{n-1}^nV_\mu \cong \bigoplus_{\lambda:\mu\vdash \lambda}V_\lambda. $$ This is proved in James' book The Representation Theory of Symmetric Groups around page 34.