I am not certain if this is going to be helpful, but I'm going to try re-expressing what the equations describe, in non-mathy words. It seems to me that the local understanding you are looking for actually is contained in Maxwell's equations, and that you are blocking on something in that description. Of course I could be quite mistaken in this; it's hard to understand a multi-year confusion based on a few forum posts. So if I'm not helpful here, sorry!

Let me start with an antenna; that is, a straight, electricity-conducting piece of metal. I run an electric current through it, or more accurately, I start running a current. As I'm doing this, the electric field within the antenna is changing; electrons are moving around, spitting out photons, and generally changing their state, heading towards the steady movement they'll have when the current is fully established. Before they can get there, however, I perversely and maliciously reverse the current's direction, and the electrons scramble to attain a completely different steady state! In fact, at no time in the following is the antenna going to be in an equilibrium state; I, the experimenter, am constantly changing its condition and keeping the electrons hopping.

Now, when I change the electric field in this manner, that change causes a magnetic field to arise. It seems possible to me that this step is the cause of your confusion, so I'll digress a bit: Why does a changing electric field cause a magnetic field? Maxwell's equations do not say anything about the causation; they merely quantify the observed fact. Studying Maxwell does not give you any greater understanding of the causation than you would have from the good old 1830s experiment of running a current through a wire and seeing a nearby compass needle deflected; all it does is to allow you to calculate how much deflection to expect. I often see this sort of confusion in the way basic physics is taught; because the equations are the full description of what's going on, people expect them also to contain the full understanding at the causal level. So there is confusion about Maxwell, and also about special relativity; people ask "What does it mean that the time-direction's sign is reversed in the inner product?", and the answer is that it describes the way matter behaves, but the causality is much deeper. It may be a mistake to teach Newton before anything else in physics, just because F=ma is so intuitively clear; we all see that this is just a formalisation of the way rocks behave. Throw the rock harder and it hits the other monkey faster, causing more damage: Our brains are well adapted to this piece of physics! But it does us a disservice in studying other equations, because we expect them to be similarly clear and contain a similar causal-level explanation, and they just don't.

At any rate, then, the crucial point is that when I move the electrons, they cause a magnetic field to exist. If you look deeply enough into QED, you can find an explanation of this in the way the force-carrier photons are moving, but personally I am quite unable to visualise this; all I can do is go through the math that shows Maxwell's equations coming out as the classical limit of QED. (Well, anyway, I could do it for an exam some years ago.) However, it may be helpful to visualise it like so: When I accelerate the electrons, the virtual photons that they spat out a few nanoseconds ago (messengers for their electric field) are 'unable to return home' (home having moved) and must find something else to do; the something else is to interact with other particles, which we measure as a magnetic field.

Now, because I'm varying the acceleration of the electrons, the size of the magnetic field caused by their acceleration is also changing. And a varying magnetic field... causes an electric field. It is inaccurate, but possibly helpful, to view this as being caused by the aforementioned no-longer-so-virtual photons interacting with electrons in the quantum foam and moving them around, creating a momentary polarisation of space.

So now there is a changing electric field not just at the antenna where I'm doing my thing with the electric current, but also some distance away where the resulting magnetic field is causing a reflection of that process. Rinse and repeat: This electric field's changing causes a magnetic field over here, which causes... You will get a chain of electric/magnetic fields running across the entire universe. This is what is meant by a 'light wave'.

Maxwell allows us to describe in numbers what I just described in words, but the causal understanding is all in the observed fact that a changing (not a steady) electric current causes a compass needle to deflect. You can take this as unadorned, experimental observation, "We don't know why that happens", or you can try to visualise it in terms of messenger photons as I outlined above. I hope that helps.