The reason to introduce these functions is clear from calculus III. We don't even need to consider anything too fancy. Consider the following:
$ \frac{\partial x}{\partial y}=0 $
How is this shown? We need to introduce a function $f(x,y) = x$ and then calculate from the definition for partial derivative with respect to $y$ at $(a,b)$
$ \frac{\partial x}{\partial y}(a,b) = \lim_{h \rightarrow 0} \frac{f(a,b+h)-f(a,b)}{h} = \lim_{h \rightarrow 0} \frac{a-a}{h} = \lim_{h \rightarrow 0} 0 = 0 $
Of course this holds for all $(a,b)$ so we can drop it without ambiguity.
Fun exercise: do the same calculation for $f(\vec{x}) = e_j \cdot \vec{x} = x_j$ to prove that $\frac{\partial x_j}{\partial x_i} = \delta_{ij}$. Here the partial derivative with respect to $x_i$ would be defined by $ \frac{\partial f}{\partial x_i}(\vec{p}) = \lim_{h \rightarrow 0}\frac{f(\vec{p}+he_j)-f(\vec{p})}{h} $
Oneil's point is that without this notation it makes many statements relating maps on the surface to maps on $\mathbb{R}^n$ awkward. Often an embedded surface can be covered by restricting the ambient maps $x,y,z$ on $\mathbb{R}^3$. For example, a plane with $z=2$ natrually inherits the $x,y$ coordinate maps by simply restricting their domain from the natural $\mathbb{R}^3$. In short, it's about having a notation for formulas you need to write.
If coordinates are abhorrent then perhaps you should read Lang's differential geometry, I believe he makes an effort to avoid unnecessary coordinated formulas. Elementary Differential Geometry by O'neill is written at a very different level. It is meant to be a natural extension from calculus III.