Let $\pi :Y\rightarrow X$ be the projection. The bundle $\pi ^*V$ has a canonical section $s$ (the diagonal, if you think of $\pi ^*V$ as $V\times _XV$), which vanishes exactly along $i(X)$. Thus $i_*\mathcal{O}_X$ has a Koszul resolution
$$ \ldots \rightarrow \wedge^2\pi ^*V^*\rightarrow \pi ^*V^* {\buildrel {s}\over {\longrightarrow}}\ \mathcal{O}_Y\rightarrow i_*\mathcal{O}_X$$
by locally free $\mathcal{O}_Y$-modules. Applying
$\ \underline{\mathrm{Hom}}_{\mathcal{O}_Y}(-,i_*\mathcal{O}_X)$ kills the differentials, so we get
$R\underline{\mathrm{Hom}}_{\mathcal{O}_Y}(i_*\mathcal{O}_X,i_*\mathcal{O}_X)\cong \oplus \
\wedge^pV[p]$, then $R\mathrm{Hom}_{\mathcal{O}_Y}(i_*\mathcal{O}_X,i_*\mathcal{O}_X)\cong \oplus R\Gamma (\wedge^pV[p])$, hence the result by applying $H^k$.