Consider that we have a set of points $S$ in the plane $\mathbb R^2$ or in the space $\mathbb R^3$ and we also consider the one to one mappings $f:S \to S$ which have the following property:
They maintain the distances (if $A,B \in S$ points, then $f(A),f(B)$ are also points $\in S$, and the distance of $f(A)$ from $f(B)$ is equal to the distance of $A$ from $B$), that is, $f$ is an isometry.
Then, the set of these $f$ consist of the symmetries of the set of points $S$.
Could you explain me the last sentence?
"symmetries" are rotations, reflections, translations, and combinations of those, that map a space onto itself.
As an example, the symmetries of the square form the group $D_8$, which consists of identity, 3 rotations, and 4 reflections.
The symmetries of $\mathbb R^2$ or $\mathbb R^3$ are the isometries of those spaces.