If you want to dive really deep into Bézier curves, I found this really detailed and interactive online book about it (https://pomax.github.io/bezierinfo/). It really helped me understand the nitty gritty.

From what I've gathered, it seems that Bézier curves are created by multiple linear interpolation. Within the line between p0 and p1, there is a movable point (Called q0). Similarly, there is one within p1 and p2 (q1), and p2 and p3 (q2). Draw a linear Bézier curve between q0 and q1, and q1 and q2. On these two lines is once again, two more moveable points (r0, r1). Finally, draw a linear Bézier curve between r0 and r1. Along this line, there is a moveable point b0 which will describe the cubic Bézier curve. I'm sorry if this was too convoluted. I think writing it out really helped me understand it's construction.

Yup that's correct. Drawing out an example by hand also helps!

I think what Spectato described is basically the de Casteljau algorithm? There is a really neat visualization in the proceeding slides!