What does it physically mean to take the curl of the curl of a field in the derivation of the electromagnetic wave equation from Maxwell's equations, as presented here, on Wikipedia? Why was it a logical first step to take?
From the point of view of electromagnetism I can see it the following non-mathematical way.
$curl$ or $rot$ differential operators calculates magnitude of "vorticity" of the field, i.e. how much it is "spinning" or changing in rotation.
Now, "vorticity" of the "vortex" is how much this "vortex" spinning again. Take, for example, long spring and connect its ends as torus. Here you will measure "vorticity" of the "vortex".
This model presumably can be applied to the $E$ and $B$ field in electromagnetism to demonstrate its interdependence and nature of EMR in individual $E$ and $B$ field components.