Are there any good references that provide an intuitive, motivated, or visual explanation of the conjugate gradient method?
A great reference for the conjugate gradient method is
The author writes in the first paragraph:
When I decided to learn the Conjugate Gradient Method (henceforth, CG), I read four different descriptions, which I shall politely not identify. I understood none of them. Most of them simply wrote down the method, then proved its properties without any intuitive explanation or hint of how anybody might have invented CG in the first place. This article was born of my frustration, with the wish that future students of CG will learn a rich and elegant algorithm, rather than a confusing mass of equations.
Not only does the author provide a clear explanation of the conjugate gradient method using diagrams, but they also provide great intuitive explanations for its convergence properties.