If $A\in\Hom(E)$ is normal, then there is an orthonormal basis $e_1,\ldots,e_n$ for $E$ and complex numbers $\l_1,\ldots,\l_n$ such that $Ae_j=\l_je_j$ and $A^*e_j=\bar\l_je_j$.

Since both $A$ and $A^*$ are diagonalizable and the family $\{A,A^*\}$ commutes, it's simultanously diagonalizable, i.e. there is a normalized basis $e_1,\ldots,e_n$ such that $Ae_j=\l_j e_j$ and $A^*e_j=\mu_j e_j$. By definition of $A^*$ it follows that $\mu_j=\la A^*e_j,e_j\ra=\la e_j,Ae_j\ra=\bar\l_j$ and for $j\neq k$: $\l_j\la e_j,e_k\ra=\la Ae_j,e_k\ra=\la e_j,A^*e_k\ra=\l_k\la e_j,e_k\ra$, i.e. $\la e_j,e_k\ra=0$ if $\l_j\neq\l_k$. Thus the eigen-spaces of $A$ form an orthogonal decomposition.