Recall that if $A,B \in \mathbb{R}^{m \times n}$ then
\begin{equation}
\langle A, B \rangle = \text{Tr}(A^T B)
\end{equation}
and
\begin{align*}
\|A\|_F^2 &= \langle A,A \rangle \\
&= \text{Tr}(A^T A) \\
&= \text{Tr}(A A^T).
\end{align*}
Let $f:\mathbb{R}^{m \times n} \to \mathbb{R}$ such that
\begin{align*}
f(X) &= \frac12 \| X A^T \|_F^2 \\
&= \frac12 \text{Tr}(X A^T A X^T).
\end{align*}
Let $J$ be the $m \times n$ matrix whose entries are all $0$ except $J_{ij}$ which is equal to $1$.
Let $\Delta X = \epsilon J$, where $\epsilon > 0$ is tiny.
Then
\begin{align*}
f(X + \Delta X) &= \frac12 \text{Tr}((X + \Delta X)A^T A (X + \Delta X)^T) \\
&= \frac12 \text{Tr}(X A^T A X^T) + \frac12 \text{Tr}(\Delta X A^T A X^T)
+ \frac12 \text{Tr}(X A^T A \Delta X^T) \\
& \qquad + \frac12 \text{Tr}(\Delta X A^T A \Delta X^T) \\
&\approx \frac12 \text{Tr}(X A^T A X^T) + \frac12 \text{Tr}(\Delta X A^T A X^T)
+ \frac12 \text{Tr}(X A^T A \Delta X^T) \\
&= \frac12 \text{Tr}(X A^T A X^T) + \text{Tr}(X A^T A \Delta X^T) \\
&= f(X) + \left\langle X A^T A,\Delta X \right\rangle \\
&= f(X) + \epsilon \left \langle X A^T A,J \right\rangle.
\end{align*}
Comparing this result with the equation
\begin{equation}
f(X + \epsilon J) \approx f(X) + \epsilon \frac{\partial f(X)}{\partial X_{ij}}
\end{equation}
we see that
\begin{equation}
\frac{\partial f(X)}{\partial X_{ij}} =
\left \langle X A^T A,J \right\rangle.
\end{equation}