Parametrize a curve with respect to arc length “It is often useful to parametrize a curve with respect to arc length because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system.” Quoted from Jim Stewart Calculus, Early Transcendentals, fourth edition, page 850
I understand why arc length does not depend on a particular coordinate system, but what does it mean to say that it “arises naturally” from the shape of a curve? 
 A: In short, this is a comment that is best understood in contexts, yet less so by itself and in general. See below.

Let's start in general.
Arclength of an arbitrary (continuous) curve $f\in\mathcal{C}([a,b],X)$ is typically defined as
$$
\text{arc length}=\sup_{\text{all partitions }\mathcal{P}}\sum_{i=1}^N||f(x_i)-f(x_{i-1})||=:L([a,b]) \quad...(1)
$$
where $\mathcal{P}=\{ a=x_0\leq\cdots\leq x_N=b\}$ and $X$ is a metric space, for example $\mathbb{R}^n$.
If you are more in the physical sciences instead of mathematics, you may interpret (1) as going along the curve in discrete and finite number of steps while summing the straight line distances traversed these steps, and the arc length is the "least upper bound", or maximum distance you obtain.
See picture below from Wikipedia.

Based on (1), you can prove that $L([a,t_1])\leq L([a, t_2])\quad\forall a\leq t_1\leq t_2\leq b$. In other words, as you travel from one end of the curve to another, the length of the curve segment you travel so far can only increase.
As a result, the map $\mathcal{l}(t)=L([a,t])$ is a valid function. If the curve is nice, $l$ may be bijective, in which case you can recover a point on the curve by its distance on the curve from origin $a$ via $f(l^{-1}(\text{distance}))$. That is to say your curve can be parametrized by its arc length.
Let me stress again: the curve has to be nice. And Stewart did say "often".
For example, a circle in $\mathbb{R}^2$ can be considered as the image of $f\in\mathcal{C}([0,2\pi],\mathbb{R}^2)$ where $f(t)=(\cos t,\sin t)$. Points on the circle can be parameterized by distance from $(1,0)$ on the circle. The conversion from distance to Cartesian coordinates is given by $f(\text{distance})$. Here $l(t)=t$. And the form of $l$ arises from the fact that $f$ traces out a unit circle.
In this context, Stewart's words "arises naturally from the shape of the curve" has good meaning.
Now, the distance function $x,y\rightarrow||x-y||$ on $X$ does not depend on the coordinate system. So $L$ is coordinate in-dependent. (To have a (local) coordinate system means to have a (local) homeomorphism to $\mathbb{R}^n$.)

At the end, let me share my idea of why definition (1) has a good physical meaning. 
Note that when we say curve, we really mean the image set under function $f$, namely the set $C=\{f(x)|x\in[a,b]\}$. Consider any finite sample $P=\{y_i\}_{i=0}^N$ of the set $C$. ie. $P\subseteq C, |P|<\infty$.
Join the sample points by "straight" lines. Then the sum lengths of these straight line segments must be smaller than or equal to the length of the curve, should the notion of length make sense. That is
$$
\sum_{i=1}^N||y_i-y_{i-1}||\leq L<\infty
$$
Therefore, this "arc length" $L$ provides an upper bound to all possible sum lengths produced by finite sampling of the curve followed by linear interpolation. Any bounded set of the reals has a least upper bound (aka supremum). Hence the definition of arc length (1).
Now the we have the definition, we recognize that supremum can be infinity and consequently $L$ can be $\infty$. So we are allowed to have curves that does not have a finite length (or depending on your taste, a well-defined length). But the use of supremum allows us to define "arc length" on any curve.
