How is the Riemann integral a special case of the Stieltjes integral?

The idea is that we define a different way of measuring the width of the rectangle. Instead of using the difference of the two coordinates, we take the difference of the images of those two coordinates under the map $\alpha$. Basically, the idea is that the $(x_i-x_{i-1})$ is to be interpreted as the width or measure of the interval, a construction which leads more generally to the Lebesgue measure of a subset of the reals. The $\alpha(x_i)-\alpha(x_{i-1})$ leads to a different measure, starting with a different assignment of "size" to an interval.

Note that when $\alpha$ is continuously differentiable, then $\alpha(x_i) -\alpha(x_{i-1})$ is equal to $\alpha'(x) (x_i - x_{i-1})$ for $x \in [x_{i-1}, x_i]$ up to higher order terms. So in this case the Stieltjes integral with respect to $\alpha$ is the Riemann integral of the same function times $\alpha'$.

To add to Akhil's answer, the Stieltjes integral made a lot more sense to me when I started thinking about it in terms of physics. Think of taking something like the integral you get to determine moment of inertia, where it's mass--a function of position--you're integrating over, and not position itself.

In Stieltjes integral you assign different importance to different parts of the set you are integrating over.

The usefulness of this would become clearer when you know the theory of Lebesgue measure, which generalizes this even further. For example, there is the Dirac measure which when used for integrating functions cares only about the value of a function at a point (typically the origin).

Understanding it in the Stieltjes form is not any harder. Moreover, it will pave the way for measure theory.