Getting $p_y(y) = p_x(g^{-1}(y)) \left| \frac{\partial{x}}{\partial{y}} \right|$ by solving $| p_y(g(x)) \ dy | = | p_x (x) \ dx |$? My textbook has a very brief section that introduces some concepts from measure theory:

Another technical detail of continuous variables relates to handling continuous random variables that are deterministic functions of one another. Suppose we have two random variables, $\mathbf{x}$ and $\mathbf{y}$, such that $\mathbf{y} = g(\mathbf{x})$, where $g$ is an invertible, continuous, differentiable transformation. One might expect that $p_y(\mathbf{y}) = p_x(g^{−1} (\mathbf{y}))$. This is actually not the case.
As a simple example, suppose we have scalar random variables $x$ and $y$. Suppose $y = \dfrac{x}{2}$ and $x \sim U(0,1)$. If we use the rule $p_y(y) = p_x(2y)$, then $p_y$ will be $0$ everywhere except the interval $\left[ 0, \dfrac{1}{2} \right]$, and it will be $1$ on this interval. This means
$$\int p_y(y) \ dy = \dfrac{1}{2},$$
which violates the definition of a probability distribution. This is a common mistake. The problem with this approach is that it fails to account for the distortion fo space introduced by the function $g$. Recall that the probability of $\mathbf{x}$ lying in an infinitesimally small region with volume $\delta \mathbf{x}$ is given by $p(\mathbf{x}) \delta \mathbf{x}$. Since $g$ can expand or contract space, the infinitesimal volume surrounding $\mathbf{x}$ in $\mathbf{x}$ space may have different volume in $\mathbf{y}$ space.
To see how to correct the problem, we return to the scalar case. We need to present the property
$$| p_y(g(x)) \ dy | = | p_x (x) \ dx |$$
Solving from this, we obtain
$$p_y(y) = p_x(g^{-1}(y)) \left| \dfrac{\partial{x}}{\partial{y}} \right|$$
or equivalently
$$p_x(x) = p_y(g(x)) \left| \dfrac{\partial{g(x)}}{\partial{x}} \right|$$

How do they get $p_y(y) = p_x(g^{-1}(y)) \left| \dfrac{\partial{x}}{\partial{y}} \right|$ or equivalently $p_x(x) = p_y(g(x)) \left| \dfrac{\partial{g(x)}}{\partial{x}} \right|$ by solving $| p_y(g(x)) \ dy | = | p_x (x) \ dx |$?
Can someone please demonstrate this and explain the steps?

This is called the method of transformations. It is detailed on this site. You need to transform a function of a random variable in order to make the CDF equal to $1$. For a demonstration. 
Suppose that $X \sim \textrm{Unif}(0,1)$  and let $Y = e^{X}$ 
Note that cdf of $X$ is given by
$$ F_{X}(x) =\begin{align}\begin{cases} 0 &  x < 0 \\ \\ x   &  0 \leq x \leq1 \\ 1  &  x > 1  \end{cases} \end{align} \tag{1}$$
Then to find the cdf of $Y$ 
$$ F_{Y}(y) = P(Y \leq y) \\ P(e^{X} \leq y) \\ = P(X \leq \ln(y)) \\ = F_{X}(\ln(y)) = \ln(y) \tag{2}$$
$$ F_{Y}(y) =\begin{align}\begin{cases} 0 & y < 1 \\ \\ \ln(y)   &  1 \leq \ln(y) \leq e \\ 1  &  x > e  \end{cases} \end{align} \tag{3}$$
To obtain the pdf we take the derivative
$$f_{Y}(y) = F_{Y}^{'}(y) = \begin{align}\begin{cases}  \frac{1}{y}   &  1 \leq \ln(y) \leq e \\ 0  &  \textrm{ otherwise} \end{cases} \end{align} \tag{4}$$
Concerning the problem above, suppose that $ X \sim \textrm{Unif}(0,1)$ and that $ Y = \frac{X}{2}$
The CDF for $X$ is the same as above. Let's look at the cdf of $Y$. We note that $R_{X} =[0,1]$ so then $ R_{Y}=[0,\frac{1}{2}]$
$$ F_{Y}(y) = P(Y \leq y) \\ P(\frac{X}{2} \leq y) \\ = P(X \leq 2y) \\ = F_{X}(2y) = 2y \tag{5}$$
You are simply taking the reciprocal. To find the pdf, we differenitate.
$$ F_{Y}(y) = \begin{align}\begin{cases} 0 & y< 0 \\ \\ 2y   &  0 \leq y \leq \frac{1}{2} \\ 1  &  y > \frac{1}{2}  \end{cases} \end{align} \tag{6}$$
to find the pdf 
$$f_{Y}(y) = F_{Y}^{'}(y) = \begin{align}\begin{cases}  2   &  0 \leq y \leq \frac{1}{2} \\ 0  &  \textrm{ otherwise} \end{cases} \end{align} \tag{7}$$
Visually the difference in the two uniform distributions can be seen below. 
$$ X\sim \textrm{Unif}(0,1)  \tag{8} $$

$$ X\sim \textrm{Unif}(0,1) , Y = \frac{X}{2} , Y \sim \textrm{Unif}(0,\frac{1}{2}) \tag{9} $$

