[proofplan]
We compute everything in a holomorphic coordinate chart. In such a chart, the Hermitian form $\theta$ is represented by a positive definite Hermitian matrix $G=(g_{j\bar{k}})$, and the induced metric on the canonical frame $dz_1\wedge\cdots\wedge dz_n$ has squared norm $(\det G)^{-1}$. The Chern curvature of a Hermitian line bundle is locally $-\partial\bar{\partial}$ of the logarithm of the squared norm of a holomorphic frame, while the Ricci form of $\theta$ is $-i\partial\bar{\partial}\log\det G$. Comparing these two local formulae gives $i\Theta_{h_\theta}(K_X)=-\operatorname{Ric}(\theta)$, and the positivity statements follow from the definitions of positivity and dual curvature.
[/proofplan]
[step:Write the Hermitian metric in a holomorphic coordinate frame]
Fix a point $p\in X$. Let $(U,z)$ be a holomorphic coordinate chart centered around $p$, where
\begin{align*}
z:U&\to z(U)\subseteq \mathbb{C}^n,\\
q&\mapsto (z_1(q),\dots,z_n(q)).
\end{align*}
On $U$, write the Hermitian form $\theta$ as
\begin{align*}
\theta=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar{z}_k,
\end{align*}
where each coefficient function
\begin{align*}
g_{j\bar{k}}:U&\to \mathbb{C}
\end{align*}
is smooth, and the matrix-valued function
\begin{align*}
G:U&\to \operatorname{Herm}_n^{+},\\
q&\mapsto \bigl(g_{j\bar{k}}(q)\bigr)_{j,k=1}^n
\end{align*}
takes values in the positive definite Hermitian matrices.
Let
\begin{align*}
e:U&\to K_X|_U,\\
q&\mapsto dz_1|_q\wedge\cdots\wedge dz_n|_q
\end{align*}
be the local holomorphic frame of $K_X$ determined by the coordinate chart. By definition of the metric on the determinant of the cotangent bundle induced from $\theta$, the squared norm of this frame is
\begin{align*}
|e|_{h_\theta}^2=(\det G)^{-1}.
\end{align*}
[guided]
We first translate the geometric objects into local coordinates. Fix $p\in X$ and choose a holomorphic chart
\begin{align*}
z:U&\to z(U)\subseteq \mathbb{C}^n,\\
q&\mapsto (z_1(q),\dots,z_n(q)).
\end{align*}
Because $\theta$ is a Hermitian $(1,1)$-form, it has the local expression
\begin{align*}
\theta=i\sum_{j,k=1}^n g_{j\bar{k}}\,dz_j\wedge d\bar{z}_k.
\end{align*}
Here each
\begin{align*}
g_{j\bar{k}}:U&\to \mathbb{C}
\end{align*}
is smooth, and the matrix
\begin{align*}
G:U&\to \operatorname{Herm}_n^{+},\\
q&\mapsto \bigl(g_{j\bar{k}}(q)\bigr)_{j,k=1}^n
\end{align*}
is positive definite Hermitian at every point. The positivity is exactly the pointwise positive definiteness of the Hermitian metric.
The canonical bundle is
\begin{align*}
K_X=\Lambda^n T^{*(1,0)}X.
\end{align*}
The chosen holomorphic coordinates give the local holomorphic frame
\begin{align*}
e:U&\to K_X|_U,\\
q&\mapsto dz_1|_q\wedge\cdots\wedge dz_n|_q.
\end{align*}
Since the metric on $K_X$ is induced by the Hermitian metric $\theta$, and $K_X$ is the determinant of the holomorphic cotangent bundle, the squared norm of the determinant cotangent frame is the reciprocal determinant of the tangent metric matrix:
\begin{align*}
|e|_{h_\theta}^2=(\det G)^{-1}.
\end{align*}
This reciprocal is the key sign source in the theorem.
[/guided]
[/step]
[step:Compute the Chern curvature of the canonical bundle]
For a Hermitian holomorphic line bundle with local holomorphic frame $e$ and squared norm $|e|_h^2$, the Chern curvature is locally
\begin{align*}
\Theta_h=-\partial\bar{\partial}\log |e|_h^2.
\end{align*}
Applying this formula to $(K_X,h_\theta)$ on $U$ gives
\begin{align*}
\Theta_{h_\theta}(K_X)
&=-\partial\bar{\partial}\log |e|_{h_\theta}^2\\
&=-\partial\bar{\partial}\log \bigl((\det G)^{-1}\bigr)\\
&=\partial\bar{\partial}\log\det G.
\end{align*}
Multiplying by $i$,
\begin{align*}
i\Theta_{h_\theta}(K_X)=i\partial\bar{\partial}\log\det G.
\end{align*}
[guided]
We now compute the curvature of the Hermitian line bundle $(K_X,h_\theta)$ in the frame $e$. For a Hermitian holomorphic line bundle with local holomorphic frame $e$ and squared norm $|e|_h^2$, the Chern curvature is
\begin{align*}
\Theta_h=-\partial\bar{\partial}\log |e|_h^2.
\end{align*}
This formula applies because $e$ is a nonvanishing holomorphic local frame and $h_\theta$ is a smooth Hermitian metric.
From the previous step,
\begin{align*}
|e|_{h_\theta}^2=(\det G)^{-1}.
\end{align*}
Substituting this into the curvature formula yields
\begin{align*}
\Theta_{h_\theta}(K_X)
&=-\partial\bar{\partial}\log |e|_{h_\theta}^2\\
&=-\partial\bar{\partial}\log \bigl((\det G)^{-1}\bigr).
\end{align*}
Since $G(q)$ is positive definite Hermitian for every $q\in U$, the function $\det G:U\to (0,\infty)$ is smooth and positive, so $\log\det G$ is a smooth real-valued function on $U$. Therefore
\begin{align*}
\log \bigl((\det G)^{-1}\bigr)=-\log\det G,
\end{align*}
and hence
\begin{align*}
\Theta_{h_\theta}(K_X)
&=-\partial\bar{\partial}\bigl(-\log\det G\bigr)\\
&=\partial\bar{\partial}\log\det G.
\end{align*}
Multiplying by $i$, we obtain
\begin{align*}
i\Theta_{h_\theta}(K_X)=i\partial\bar{\partial}\log\det G.
\end{align*}
[/guided]
[/step]
[step:Compare the curvature formula with the Ricci form]
The Ricci form of the Hermitian metric $\theta$ is locally defined by
\begin{align*}
\operatorname{Ric}(\theta)=-i\partial\bar{\partial}\log\det G.
\end{align*}
The curvature computation gives
\begin{align*}
i\Theta_{h_\theta}(K_X)
=i\partial\bar{\partial}\log\det G
=-\operatorname{Ric}(\theta)
\end{align*}
on $U$. Since $p\in X$ was arbitrary and both sides are globally defined real $(1,1)$-forms, the identity holds on all of $X$:
\begin{align*}
i\Theta_{h_\theta}(K_X)=-\operatorname{Ric}(\theta).
\end{align*}
[/step]
[step:Deduce positivity for the canonical and anticanonical bundles]
A Hermitian holomorphic line bundle $(L,h)$ is positive exactly when the real $(1,1)$-form $i\Theta_h(L)$ is positive definite. If $\operatorname{Ric}(\theta)$ is negative definite, then $-\operatorname{Ric}(\theta)$ is positive definite, and the identity
\begin{align*}
i\Theta_{h_\theta}(K_X)=-\operatorname{Ric}(\theta)
\end{align*}
shows that $(K_X,h_\theta)$ is positive.
For the dual line bundle $K_X^{-1}$ equipped with the dual metric $h_\theta^{-1}$, the Chern curvature satisfies
\begin{align*}
\Theta_{h_\theta^{-1}}(K_X^{-1})=-\Theta_{h_\theta}(K_X).
\end{align*}
Therefore
\begin{align*}
i\Theta_{h_\theta^{-1}}(K_X^{-1})
&=-i\Theta_{h_\theta}(K_X)\\
&=\operatorname{Ric}(\theta).
\end{align*}
If $\operatorname{Ric}(\theta)$ is positive definite, this is a positive definite real $(1,1)$-form, so $(K_X^{-1},h_\theta^{-1})$ is positive.
[/step]
