I'm trying to understand why there's f(a) and f(b), why is b being subtracted from x, why it's divided by a-b, why there's a similar thing for f(a), why the two are being added, why the zero values are important, and what the whole thing actually means.

It’s simpler if you use 0 and 1 instead of a and b. If we want a linear function such that

f(0) = A, f(1) = B,

then the unique solution is

f(t) = (1-t)A + tB

The more general version with a and b is just composing this with a transformation of [a, b] into [0, 1]

Relevant keywords: linear interpolation, affine combination, barycentric coordinates.

It's demonstrating how to find the line that passes through two points x=a, x=b on the function y=f(x).

Without having read the actual page, I assume he's going to move a and b close to eachother to approximate the tangent of the function at a point.

edit: In higher level math, you use circles/spheres or parabolas/paraboloids to approximate functions, but in high school level calculus you stick to using a straight line to approximate a function

Unfortunately, he doesn't do that. There's an even more complicated formula under that which is supposed to be related to the slope of some line and a ratio between b and a (I think?) and then I'm guessing he reduces it, but doesn't explain how he did it.

I've tried learning calculus 3-4 times during my life, using materials for an "absolute beginner", and this has always been my experience, as if one were teaching programming by going from "This is a variable. It can store data." directly to "A monad is a monoid in the category of endofunctors." To this day I have no idea what calculus even is, or what it's for.

You want a linear function f(x) such that f(a)=A and f(b)=B given a, b, A, B. A valid f(x) (actually it's the only one) is defined as

             x-b     x-a
    f(x) = A --- + B ---
             a-b     b-a

You can calculate f(a) and f(b) to confirm that it works.

However, I don't find this kind of guide any good for learning math...