This paragraph is taken from Supergravity book by Freedman and Van Proeyen.he simplest objects to define on a manifold $M$ are scalar functions $f$ that map $M \rightarrow \mathbb{R}$.
We say that the point $p$ maps to $f(p)$ = z $\in \mathbb{R}$. On each coordinate patch $M_i$ we can define the compound map $f\circ \phi_i^{-1}$ from $\mathbb{R}^D \rightarrow \mathbb{R}$ as $f_i(x) = f\circ \phi_i^{-1}(x)=z,$ where $x$ stands for {${x^{\mu}}$}. On the overlap $M_i \cap M_j$ of two patches with local coordinates $x^{\mu}$ and $x'^{\nu}$ pf the point $p$, the two descriptions of $f$ must agree. Thus $f_i(x)=f_j(x')$.
To me this is a little bit difficult to imagine and thus comprehend without figures, because I am new to this formal way of illustrating things. I am of course not asking for figures.
My question is why was it said that scalar functions are objects on $M$ that "map" $M\rightarrow \mathbb{R}$. I thought objects are not the ones that map but rather charts are the ones that map.
Another thing is about the second paragraph above, if it could be said in any more informal way I would truly appreciate it.
Mathematicians call lots of things maps, but this concept is really no different from a scalar field. Given a point (which is an element of $M$), the scalar function returns a number (an element of $\mathbb R$).
A chart maps an open set $U \to \mathbb R^n$--for each point on $U$, there is a corresponding point in $\mathbb R^n$. That's different from $U \to \mathbb R$, which merely says that each point corresponds to some number. Moreover, the requirements of a chart insist that the chart be one-to-one, while several points on $U$ could correspond to the same number under a scalar field.
A more casual way of describing your second paragraph is like so: given a scalar field on $M$, we can use a coordinate chart to define a corresponding scalar field on $\mathbb R^n$.
Since the scalar field should have a single value at any point on $M$, when multiple charts are used, these can produce scalar fields on different regions of $\mathbb R^n$. These fields must be equal to each other under the change of chart (coordinates).