I've been trying to understand the same thing: what those equations actually describe. So far I've come up with this analogy. The B field describes surface of the magnetic medium or substance of unknown origin, the E field is merely the apparent velocity of this surface. Electrons and other charged particles blindly run down the slopes of the magnetic medium. There's also this rot B and rot E part that creates vorticity, similar to what we see in liquids.
This sounds very much like the aether hypothesis of the 19th century which was abandoned since it is incompatible with many experimental observations, such as the Michelson-Morley experiment. The modern point of view is that EM fields are fundamental and have no underlying mechanism such as an aether medium or something like that.
Sure, there are some loopholes like a dragged aether that could resolve the contradiction with Michelson-Morley. But you are going to have a very hard time explaining all the other tests of special relativity such as Ives-Stilwell type experiments.
