I'll try to elaborate on what the book is saying. There are many potential ways to define the notion of a function $f : X \to Y$. Whichever definition you give, it is possible to define the set $$\mathrm{Gr}(f) = \{ (x,f(x)) \mid x \in X \} \subseteq X \times Y$$ No matter what definition of 'function' you take, this set has the property that, for all $x \in X$, there is a unique $y \in Y$ such that $(x,y) \in \mathrm{Gr}(f)$; in particular, for a given value of $x \in X$, this unique value of $y$ is precisely $f(x)$.
Suppose now that $G \subseteq X \times Y$ is an arbitrary subset such that, for all $x \in X$, there is a unique $y \in Y$ such that $(x,y) \in G$. Then $G=\mathrm{Gr}(f)$ for a unique function $f : X \to Y$, since $f$ is determined by letting $f(x)$ be equal to the unique element of $y$ for which $(x,y) \in G$.
Thus there is a bijective correspondence between
- Functions $f : X \to Y$; and
- Subsets $G \subseteq X \times Y$ such that, for all $x \in X$, there is a unique $y \in Y$ with $(x,y) \in G$.
What the author then (implicitly) claims is that in light of this observation, we can simply say that a function $f : X \to Y$ is a subset of $f \subseteq X \times Y$ satisfying the above condition. Given $x \in X$, you can then recover the value $f(x)$ as being the second component of the only pair in the subset whose first component is $x$.
It's not a circular definition, but it does mean that $f=\mathrm{Gr}(f)$.
Side-note: I have reasons for disliking the convention of identifying functions with their graphs, but they're outside the scope of your question, so I'll refrain from ranting.
