Mechanism by which electric and magnetic fields interrelate

There isn't a mechanism. You're trying to find a mechanism for how two abstract objects can exchange identities. Any mechanism involving abstractions must consist of abstractions. So,the only way to explain it is through mathematics.

Least abstract way to look at it

I feel that the least abstract way to explain it is to look at two stationary charges. They attract via the Coulomb (electrostatic) force. Now run perpenducular to the line joining their centers. Each charge creates a magnetic field as it is moving (moving charge can be thought of in certain cases as current). The magnetic field acts upon the other charge, creating a force. Meanwhile, the electric force has decreased (no longer electtoSTATIC). The net force is the same, but part of it is magnetically caused.

Relativistic way

Another way to look at it is to remember that EM fields are set up by EM radiation. An EM wave carries oscillating EM fields with it; see pictures here. A point charge radiates EM waves in all directions. The oscillating E field of one of these waves interferes with the E field from a nearby wave constructively, creating a nearly non-oscillating field, which decreases as distance squared (Comes from the fact that intensity of a point source $\propto 1/r^2$), giving us Coulomb's law. The oscillating magnetic fields destructively interfere, so we get no net magnetic field.

Now, if you start moving with respect to the charge, things get interesting. The relativistic doppler effect will act upon the EM waves, altering them (since the speed of light is the same in all frames, we can't apply relative velocities to it). The interferences won't work quite the same, and we'll get a bit of a magnetic field and mainly an electric field. Move faster, and the magnetic field intensity increases, E decreases. Accelerate, and you get complicated stuff. Note that infact em waves are radiated only by an accelerated charge. A sitting charge does not emit em waves. The waves emitted by an accelerated charge produce change in the fields. The easiest way to visualize this is by assuming that the em waves are radiated in all cases.

I think that explains it without too many abstractions..

The simplest explanation I know of requires only one test charge and two reference frames with a relative velocity between them.

Frame 1: The charge is at rest. It is the source of a (purely) electric field.

Frame 2: The charge is moving. It is a current, and the source of a magnetic field.