What is fundamentally happening that causes light to change its orientation when repeatedly polarized? When light is passed through two polarizers successively, its intensity and orientation afterwards depends on the angle between the polarizers and the orientation of the most recent polarizer, respectively.
Why is this the case? What is physically happening, in the most fundamental terms possible, between the electric field and the polarizers that makes the light oscillation change its orientation to that of the newest polarizer, and why is the new intensity governed by Malus's law?
Maybe to help communicate the intent behind my question, my original question was this:

Why is light not fully blocked by one vertical polarizer, one
  polarizer at 45° from the x axis and one horizontal polarizer, set up
  in that order?
By my mental model, only vertically oriented light passes through the
  first, then, of that light, only a limited amount (half, by Malus's
  law, though I don't understand how this works; see the paragraph
  below) should make it through the 45° rotated polarizer. Why is this
  light not still vertically polarized and unable to propagate through
  the horizontal polarizer?
I have trouble with the fact that any makes it through the 45° rotated
  polarizer at all. My mental construction of what light waves are must
  be quite flawed. I visualize light such that, were its oscillations of
  macroscopic size, they could be completely hidden behind a vertically
  oriented pencil. Trying to drop a pencil through a slit in a cardboard
  box would only succeed were the box slit of the exact orientation as
  the pencil. Could someone please explain to me the correct mechanics
  or give me a better analogy of this situation?

I then encountered this question. From it I can see what happens, but I don't understand why the polarizer makes light behave that way.
After reading the Wikipedia page on polarizers, as suggested by ACuriousMind, below is my current guess as to what's happening. Is this correct/close?

Instead of a box with a slit, imagine a sieve of wire with one component of the mesh removed, so every strand of wire is aligned (and all the perpendicular strands are gone). 'Passes through' is bad terminology: the light that propagates on the other side of the polarizer accompanies the magnetic field that arises from the movement of the electrons in each wire. Everywhere on this wire half sieve that the 'pencil' touches, electrons in the wire will vibrate with the same orientation.
If the incident polarized light is aligned with the wires in the sieve, the electrons in the wire will be able to move the greatest amount, moving along each wire's length. If it is perpendicular to the wires, electrons will only be able to move the width of the wire; the oscillations in the electric field this causes on the other side of the sieve/polarizer will therefore be minimal. If the light hits the sieve at an angle like 45°, the electrons will be able to move further than the horizontal width of the wire, so the intensity will be greater.
There must be some factor, a resistance in the wire, that limits electron movements, as the following diagram is clearly not correct for the red induced movement when the incident light is aligned. The width of the wire equivalent in the polarizers must be such that it limits electron movement much more than the wire's resistivity does (or the intensity left from perpendicular incident light wouldn't be much less than incident light at 45°).

 A: At the fundamental level a beam of light is composed out of a huge number of photons with energy=h*nu  . The photon is an elementary particle and obeys quantum mechanical equations. A classical beam, a solution of Maxwell's equations, emerges from the synergy of the wavefunctions of all those photons, here is a simple example:


Left and right handed circular polarization, and their associate angular momenta.

As you can see in this image the photons which have each spin either +1 or -1 orient themselves to give the circular polarization of a classical wave. This happens because the wavefunction that describes a photon has several components one of which is connected to polarization. 
The wave function of each photon is described by a complex number, there exists an amplitude whose square gives the probability of finding the photon at (x,y,z) at time t, and a given phase . In an ensemble of photons the phases will build up the electric and magnetic fields that are seen macroscopically.
Polarisation of the classical light means that the electric and magnetic fields are built up in a specific way, linear or circular. An innumerable number of photons contribute to the build up . Each individual photon will have its spin either along the direction of motion or against it, the synergistically built up electric field which defines macroscopic polarization is not a simple addition.
When a light beam hits a polarizer each individual photon is filtered through according to the fields of the polarizer in such a way that a linear polarization is built up in the case you discuss. It is the behavior of the individual photons that is important, whether they will be absorbed or deflected outside the beam , so that the ones left in the beam display in synergy the new polarization direction,  that is the way to look at this. Polarizers allow photons whose wavefunction  agrees with the axis of polarization. That is what is happening at a fundamental level. 
When one is dealing with optics, it is not necessary to go through all the mathematics of second quantization in order to study macroscopic effects. Classical electromagnetism and optics works very accurately as the classical theory can be shown to be consistent with the quantum mechanical.
This is not my field of expertise but I am sure that Malus' law describes what is happening to the number of photons with respect to the orientation, and should be derivable from basic photon interactions with matter. 
A: Are you asking why describing an arbitrary polarization can be described as a combination of linear polarizations (not necessarily in phase),  or what the physical mechanism is within the polarizing material?
The answer to the  first case is relatively simple: any polarizer will block the component of incoming light perpendicular to its axis and pass the other axis. Since any vector can be decomposed into 2 perpendicular components, you can  repeat this process, albeit with loss of total signal, ad infinitum.
The specific mechanism of polarization is related to nonhomogeneous structure in the medium, which could be stressed molecules (typical plastic polarizing sheet), asymmetric crystals, and so on.
A: Whenever someone investigates the interaction between photons and edges we interpret fringes behind an edge as manifestation of particles wave character. And at the same moment we always emphasize that this waves are not observable direct. So it's only one of the possible interpretations that from fringes with a wavelike intensity distribution behind an edge we can conclude about a wave-particle-duality.
To interpret the fringes as a result of the interaction between photons (or electrons) and the electric field of the electrons of the edges surface is not common but has some charme. No more need in intepretation of self interference of an electron (or photon) with itself in single particle experiments. No more sentences like "we can mathematically write down but not describe what happens".
The common field between photon and electrons is quantized and fringes are the manifestation of this quantized field. This has to be proofed for example by changing the electric potential of the edges.
As any interaction is related at the end to surface electrons or an electric or magnetic fields by the wave-particle-duality one describe this interactions (Schrödinger, Dirac). And trying to inspect this interaction one need other photons or some other field. This disturbs the common field between edge and particle and the fringes disappear.
It is not exact to say that closing one of the slits in a double slit experiment fringes disappear at all. Every edge produces fringes and in the case of one slit ore in the case of an edge one see fringes too.
A well designed polarizer let approx. 50% of the light through. The polarization direction of the incoming photons is usually equal distributed. Most of the 50% of the incoming photons is turned than by the electric field from the surface electrons. The polarized photons have than no chance to go through the 90° polarizer. The evidence for the interaction between photon and the edges is the polarizer under 45° between the two twisted polarizers.
P.S. This picture shows the influence of a changing electrical field to fringes made by electrons.
Here you see how this experiment with electrons was arranged in generally.
