If $F:N \to M$ is a $\mathscr{C}^{k \geq 1}$-submersion and $S \subset N$ is a $\mathscr{C}^k$-submanifold, then its preimage $F^{-1}(S)$ is a $\mathscr{C}^k$ submanifold of M.
My ideas so far:
It is clear that $F^{-1}(S)$, equipped with the subspace-topology of M is Hausdorff and second-countable. So all that is left is to construct a $\mathscr{C}^k$-atlas.
Let $p \in F^{-1}(S)$ be an arbitrary point.\
Since F is a submersion, by the constant-rank theorem there exist charts $\phi:U \to \phi(U) \subset \mathbb{R}^m$ where $p \in U$ is an open neighbourhood of p and $\psi:V \to \psi(V) \subset \mathbb{R}^n$ with V an open neighbourhood of $F(p)$ such that $\psi \circ F \circ \phi^{-1}:\phi(U) \to \psi(V)$ becomes a projection $(x_1,...,x_m) \mapsto (x_1,...,x_n)$.
Since S is a submanifold of N, there exists a slice-chart $\sigma:W \to \psi(W) \subset \mathbb{R}^n$ such that $\sigma(W \cap N) = (\mathbb{R}^{k}\times\{0\}^{n-k})\cap \sigma(W)$
Now a hint was to consider the map (($\sigma \circ \psi^{-1})\times Id_{\mathbb{R}^k})\circ\phi$. But I don't quite see why this is a submanifold chart.
Any help would be appreciated.
This looks like a homework question to me. In which case it is really important to think through the problem yourself.
So I'll give you a resource you can use. Lee's Introduction to Smooth manifolds Chaters 4 and 5 (Second edition 2013 printing) will tell you all you need. Lee talks about slice charts, the rank theorem, implicit functions, all that good stuff and how the jigsaw puzzles fit.
Getting access to Lee's book might be harder, but where there is a will there are websites. Happy hunting.