Online and in textbooks, the chain rule is usually explained like this. In the case where you're differentiating $f(g(x))$, they will define $y = f(u)$ and $u = g(x)$, then explain that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$, which is obviously true to me and very easy to understand.
The chain rule is $(g \circ f)' = (g' \circ f) \cdot f'$. It's pretty obvious that the factor $(g' ∘ f)$ is supposed to be equivalent to $\frac{dy}{du}$ above, but I don't understand how. Intuitively, it feels like $\frac{dy}{du}$ would be equivalent to $g'$ because it's rate at which $y$ is changing for each input unit of $u$.
Why does it take $u$'s rate of change into account?
From your definitions
$$y = f(u)$$ $$u = g(x)$$
then we have
$$\frac{dy}{du} = f'(u)$$ $$\frac{du}{dx} = g'(x)$$
The differential of $g$ (i.e. $\frac{du}{dx}$) is evaluated at $f(x)$, and is written as $(g'\circ f)$.