Understanding B-Splines

So recently I've been trying to create a few line drawing functions, the Bezier Curve was straightforward-ish and now I am trying to implement the B-Spline. I'm following the pdf below where it states you can get the line to pass through the control points, by using Interpolatory interval spline curves, but I'm getting a bit confused with the algebra. I'm just wondering if anyone has an resources or could explain this to me like the idiot I am?

Thanks.

https://people.csail.mit.edu/sarasu/pub/cgim02/cgim02.pdf