A real explanation of the electromagnetic sine dependency Can someone fill the gaps I have regarding what exactly changes from one state to another during a particular electromagnetic circle?

First, let's assume that this is a pure magnetic field characteristic. Starting from point $0$ it begins gradually to increase its magnitude until it eventually reaches its maximum, then starts to decrease in value and reaches a point $0$ or half circle ($180$ degrees) and this is where my confusion takes place. How come the magnetic field can have a negative value? It just doesn't makes sense to me, something either does not exist (point $0$) or does exist (somewhere above $0$).
My confusion gets even worse when it comes to comprehend the electromagnetic dependency.

Two separate fields perpendicular to each other start at point $0$, gain some magnitude over time and reach their amplitudes, then start to decrease in value and reach the half circle point $0$ ($180$ degrees), Then they both pass to their second half circle (below $0$) up to $360$ degrees.
So, what I am really asking for is general explanation about what is happening. I am aware of the mathematical point but I lack real, essential understanding of how this actually works in reality. We take many classes full of formulas, diagrams and regular exams but I feel that I am missing the bigger picture. As Richard Feynman once said, there is a big difference between knowing something (memorizing) and actually understanding it, and I feel that I merely know this, rather than understand it. I am looking here for a deep fundamental understanding, not a textbook definition.
 A: Let's recall the basic experimental fact: EM fields shove electric charge. Something about the interaction between the charge and the space (and time) around it is shoving on the charge, and electromagnetic field oscillations are simply a time-varying wobble in whatever it is that is shoving the charge. You can't get more intuitive or deeper than that, that's pretty much an overview of all we know at an everyday level about this interaction. Negative shoves are simply those shoving in the direction opposite to an arbitrarily chosen reference direction.
We then try to guess what rules quantify that shove, and see how they fit into experiment. A good beginning guess is that the rules depend on the motion state of the charge concerned: see a comb rubbed on wool draw initially at rest dust to it - evidently there is a force when the charge is still. This is the electric part. See moving charges follow circles in the neighborhood of a magnet brought near a cathode ray tube: evidently there is a force at right angles to the motion of the charge. This is the magnetic part. But it turns out from special relativity that, if we guess that the relationship is a linear function of the motion state and if it has to be independent of an inertial observer's motion state, that these experimental observations are exactly the most general kind of interaction there can be between a "field" (whatever it is about spacetime that does the shoving) and a point charge (I cite my answer here, although you probably won't grasp it until you study special relativity). So the electric and magnetic part cannot be separated and both belong to one, geometric object. They don't cause one another, they are different "bits" of a unified object.
Again from special relativity, we know that physical processes cannot travel through spacetime at faster than the universal signalling speed limit $c$. This we know from the very everyday observation that causes of things always seem to come before the effects they give rise to (see my answer here). This implies "wavelike" transmission of "shove" through space with a finite velocity less than or equal to $c$. It turns out experimentally that this velocity is $c$ - either way, the unified electromagnetic "shove" object I speak of above has to fulfill some kind of  equation that makes disturbances in it travel at a finite speed.  
A: The best way to physically understand what is going on here in my opinion is to imagine that you have fixed a positive charge fixed to live on a line and then subject it to an oscillating E field (for simplicity).  The oscillating E field will exert a force on the charge such that it will oscillate up and down (please ignore things like radiation here).  If the electric field could not attain a "negative value" then the charge would just accelerate upwards and that is not what happens in reality--in reality when we shine light on charged particles they oscillate up and down.
A: You could ask the same of water waves. When displacement is 0 the momentum and kinetic energy of the water is maximum, and this will drive the subsequent displacement.
For the electromagnetic wave energy of the E and B field both dE/dt and dB/dt are maximum when E and B are zero. In the same way as for physical displacement, these fields can't instantaineously without interaction drop their first differential (rate of change) to zero (which would represent loss of all their energy). 
I wouldn't go beyond that and start to look a particle or quantum models unless you want to switch courses. Just to know that it is something that can retain it's dE/dt & dB/dt as well as an E and B field, I hope that will satisfy you.
