Understanding the derivation of Axis-Angle matrix visually

I'm familiar with the rotation matrix and its three-angle parametrizations (i.e. Euler's angles). I'm stumbling now on axis-angle representation. In the book I'm reading, it says

One approach is to note that the rotational transformation $R = R_{z,α}R_{y,β}$ will bring the world z-axis into alignment with the vector $k$. Therefore, a rotation about the axis $k$ can be computed using a similarity transformation as

$$ \begin{align*} R_{k,\theta} &= RR_{z,α}R^{-1} \\ &= R_{z,α}R_{y,β} R_{z,\theta} R_{y,-β}R_{z,-α} \end{align*} $$

I'm not able to visually understand why the reverse of the first sequence of rotations is carried out.

The general process works like this - rotate $k$ to align with the $z$-axis, then rotation around the $z$-axis, rotate $k$ back to its original position. Since you are familiar with Euler angles I'm sure you know that rotation do not commute in three dimensions so it's important to undo them in the right order. You'll often see transformations of the form $XAX^{-1}$ in non-commutative settings such as this.