Intuition behind using the inverse of a Hessian matrix for automatically estimating the learning rate (aggression parameter) in gradient descent.
It's not an intuition, it's mathematics. And more precisely, the quadratic expansion of a function.

Say you have $f(x)=f(x_n+\Delta x)=f(x_n)+f'(x_n)\Delta x + f''(x_n)\Delta x^2$ and you want to get to a point where $f'(x) = 0$, then you will use $\Delta x=-\frac{f'(x_n)}{f''(x_n)}$.

Generalizing to n dimension, $f''$ is the Hessian of the equation.

So of course this works perfectly for least squares, it will only "work" close to the solution for non quadratic functions.