I see two ways of proving this. The first is a very quick proof, if you know the following result about the relationship between the minimal polynomial and the Jordan canonical form (this is a useful result to know, but perhaps not so obvious to prove):
Theorem $1$: Let $\lambda_1, \dots, \lambda_k$ be all the distinct eigenvalues of the linear transformation $A: E \to E$. For each $i$, let $n_i$ be the maximum size of the Jordan blocks corresponding the eigenvalue $\lambda_i$. Then, the minimal polynomial of $A$ is
\begin{equation}
m_A(t) = \prod_{i=1}^k (t-\lambda_i)^{n_i}.
\end{equation}
(FYI: the number $n_i$ is also the nilpotency index of $(A-\lambda_iI)$ when restricted to the generalised eigenspace corresponding to $\lambda_i$.)
In your question, all the $n_i$'s are $1$ by assumption, so by Theorem $1$, the maximum size of the Jordan block corresponding to $\lambda_i$ is $1$. It follows that all the Jordan blocks are $1 \times 1$. This is equivalent to $A$ being diagonalizable.
The second proof uses the following lemma about polynomials of operators and the relationship between their kernels:
Lemma: Suppose $E$ is a vector space over a field $\mathbb{F}$, and let $f_1(t), \dots, f_k(t)$ be pairwise coprime polynomials with coefficients in $\mathbb{F}$. Then, for any linear map $A: E \to E$, we have that
\begin{equation}
\text{ker}\left( f_1(A) \circ \dots \circ f_k(A)\right) = \bigoplus_{i=1}^k \text{ker}f_i(A),
\end{equation}
i.e \begin{equation}
\text{ker} \left( \prod_{i=1}^k f_i(A) \right) = \bigoplus_{i=1}^k \text{ker}f_i(A)
\end{equation}
The proof of the lemma is by induction on $k$, which you should definitely attempt. To apply this, we take $f_i(t) = t-\lambda_i$. Then the lemma says
\begin{equation}
\text{ker}\left( m_A(A) \right) = \bigoplus_{i=1}^k \text{ker}(A-\lambda_iI).
\end{equation}
Since $m_A(t)$ is the minimal polynomial of $A$, we have $m_A(A) = 0$; i.e the kernel is all of $E$. Also, note that $\text{ker}(A-\lambda_iI)$ is precisely the eigenspace $E_{\lambda_i}$ of $A$ correpsonding to $\lambda_i$. Hence, we have shown that
\begin{equation}
E = \bigoplus_{i=1}^k E_{\lambda_i}.
\end{equation}
Recall that $A$ is diagonalizable if and only if we have such a direct sum decomposition; hence this completes the proof.
