By the construction of a manifold, for each point \$$s \in M\$$, there exists a "chart" which is a one-to-one mapping between a neighborhood around \$$s\$$ in \$$M\$$ and an open subset of \$$\mathbb{R}^n\$$, where \$$n\$$ is the dimension of the manifold.

Footnote: it's more than just one-to-one. A manifold \$$M\$$ is not just a set, but a topological space, and the charts are topological isomorphisms (aka homeomorphisms) - continuous functions with continuous inverses.

The chart establishes a strong and "smooth" correspondence between the neighborhood in \$$M\$$ and the neighborhood in the parameter space \$$\mathbb{R}^n\$$.