Let $M$ be compact smooth manifold and $G$ -- a group acting freely on $M$. The problem is to prove that there is the smooth structure on $M/G$ which makes the map $\pi: M \to M/G$ smooth.
I know only how to put topology on $M/G$ and have no idea how to define charts.
I think the idea is more or less the following. Take a small open set $U\subset M$, then the tangents to the orbits define a distribution $N\subseteq TU$. You should be able to pick a complement to $N$ in $TU$ (for example fixing a Riemannian metric on $M$ and taking the orthogonal complement with respect to it), and this complement will heuristically be $T(M/G)$. You prove that this complement is an integrable distribution, and then by Frobenius' theorem you can integrate it to a submanifold which is locally like $M/G$, and thus gives you a chart.
See e.g. around Definition 2.12 here (and the references therein).