A subset $S$ of a smooth $m$-manifold $M$ is called regular submanifold of codimension $k$ if for every $p\in S$, there is a coordinate neighborhood $(U, \phi)= (U, x^1, \dots, x^m)$ in the atlas of $M$ such that $x^1 = \dots = x^k =0$ on $U \cap S$.
Then, there are theorems like:
Let $g: M\to \mathbb{R}$ be a $C^\infty$-function. Then a non-empty regular level set $S=g^{-1}(c)$ is a regular submanifold of $M$ of codimension $1$.
Question: (From someone ignorant about differential geometry!) What is the practical use of knowing that a certain subset $S\subseteq M$ is a regular submanifold? We know it is a manifold in some natural way, but do we actually understand its structure? It is my understanding that the charts of such a submanifold are constructed using abstract results like the inverse function theorem, so we don't really understand how an atlas of this submanifold looks like. Say, for example, we are given a regular submanifold $S\subseteq M$ which we know is a submanifold by the above theorem. We have a function $S \to N$ where $N$ is some other manifold. We don't have an explicit atlas for $S$, so how do we for example show that the function $S\to N$ is smooth?