The definition of $k$-th order jet as an equivalence class $[f]_x^k$ of a function $f\in C^\infty M$ at point $x\in M$, gives you a natural map
\begin{align}
\mathcal{j}^k\colon C^\infty M &\to \mathscr{J}^k M\\
f &\mapsto (x\mapsto [f]_x^k),
\end{align}
which may be shown to be the universal differential operator of order $\leq k$ out of $C^\infty M$. This means that any differential operator of order $\leq k$ out of $C^\infty M$, say $\Delta\colon C^\infty M \to P$, with $P$ a $C^\infty M$-module, factors uniquely through $\mathcal{j}^k$ via a $C^\infty M$-linear map $\phi_\Delta\colon \mathscr{J}^k M \to P$. This implies that $D^k M$ (i.e. the module differential operators from $C^\infty M$ to $C^\infty M$ of order $\leq k$, with its left $C^\infty M$-module structure) is naturally isomorphic to $\text{Hom}_{C^\infty M}(\mathscr{J}^k M,C^\infty M)=(\mathscr{J}^k M)^*$, where $P^*$ denotes the $C^\infty M$ dual. Using this we get
$$
\text{Hom}_{C^\infty M}(D^k M,C^\infty M)= \text{Hom}_{C^\infty M}((\mathscr{J}^k M)^*,C^\infty M) = ((\mathscr{J}^k M)^*)^*=\mathscr{J}^k M
$$
where the last equality follows from the fact that $\mathscr{J}^k M$ is a finitely generated projective $C^\infty M$-module when $M$ is a smooth manifold. Passing to the limit with $k\to \infty$ gives you the isomorphism in your question.
A reference for this might be Krasil'shchik, Verbovetsky, Homological Methods in Equations of Mathematical Physics.
