What are the “standard coordinate functions” on $R^n$?

I'm reading page 44 of Differential Topology by Guillemin.

I am trying to understand this proof:

Suppose that x is any point in X (a manifold sitting in $R^n$. and that $x_1,....x_n$ are the standard coordinate functions on $R^n$ (So...is just the coordinate of $R^n$??? Just as (x,y) is a coordinate function for $R^2$?)

Then the restriction of some k of these coordinate functions $x_{i_1}.....x_{i_k}$ to X constitute a coordinate system in a neighborhood of x.

I have, absolutely no idea what they are talking about, restricting the standard basis to a manifold? how?

If anyone could enlighten me it would be great...

For $1\le i\le n$, the $i$th standard coordinate function on $\Bbb R^n$ is the map $x_i\colon\Bbb R^n\to \Bbb R$ given by $(\xi_1,\ldots,\xi_n)\mapsto \xi_i$.