Let $SO(3)$ be the Lie group of 3D rotations. Rotation about z-axis by an angle $\phi$ is represented in standard basis by this matrix:
$$ \begin{pmatrix} \cos \phi & -\sin\phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Differentiating this matrix at $\phi = 0$ we get an infinitesimal generator of rotation about z-axis: $$ Z = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$
In similar fashion we get matrices $X$ and $Y$ corresponding to rotations about x- and y-axis. These matrices are elements of $\mathfrak{so}(3)$, Lie algebra of $SO(3)$ and bracketing gives us:
$$[X, Y] = -Z, [Y, Z] = -X, [Z, X] = -Y$$
I'm not really good at visualizing things, but I'm curious is there a nice visual explanation for this?
$[X,Y]$ refers to the rate of change of $Y$ over $X$. So it is equal to $Z$. It appears to be the cross product, as @Will Jagy said. If you want a more sophicated explanation of the whole thing, I will have to start over and take some time.