The main important idea of Feynman Wheeler theory is to use propagators which are non-causal, that can go forward and backward in time. This makes no sense in the Hamiltonian framework, since the backward in time business requires a formalism that is not rigidly stepping from timestep to timestep. Once you give up on a Hamiltonian, you can also ask that the formalism be manifestly relativistically invariant. This led Feynman to the Lagrangian formalism, and the path integral.
The only reason the Feynman Wheeler idea doesn't work is simply because of the arbitrary idea that an electron doesn't act on itself, and this is silly. Why can't an electron emit and later absorb the same photon? Forbidding this is ridiculous, and creates a nonsense theory. This is why Feynman says he abandons the theory. But this was the motivating idea--- to get rid of the classical infinity by forbidding self-interaction. But the result was much deeper than the motivating idea.
Feynman never abandons the non-causal propagator, this is essential to the invariant particle picture that he creates later. But later, he makes a similar non-causal propagator for electrons, and figures out how to couple the quantum electrons to the photon without using local fields explicitly, beyond getting the classical limit right. This is a major tour-de-force, since he is essentially deriving QED from the requirement of relativistic invariance, unitarity, the spin of the photon and electron, plus gauge-invariance/minimal coupling (what we would call today the requirement of renormalizability). These arguments have been streamlined and extended since by Weiberg, you derive a quantum field theory from unitarity, relativistic invariance, plus a postulate on a small number of fundamental particles with a given spin<1.
In Feynman's full modern formalism, the propagators still go forward and backward in time just like the photon in Wheeler-Feynman, the antiparticle goes backward, and the particle forward (the photon is its own antiparticle). The original motivation for these discoveries is glossed over by Feynman a little, they come from Wheeler's focus on the S-matrix as the correct physical observable. Wheeler discovered the S-matrix in 1938, and always emphasized S-matrix centered computations. Feynman never was so gung-ho on S-matrix, and became an advocate of Schwinger style local fields, once he understood that the particle and field picture are complementary. He felt that the focus on S-matrix made him work much harder than he had to, he could have gotten the same results much easier (as Schwinger and Dyson did) using the extra physics of local fields.
So the only part of Wheeler-Feynman that Feynman abandoned is the idea that particles don't interact with themselves. Other than that, the Feynman formalism for QED is pretty much mathematically identical to the Wheeler-Feynman formalism for classical electrodynamics, except greatly expanded and correctly quantum. If Feynman hadn't started with backward in time propagation, it isn't clear the rest would have been so easy to formulate. The mathematical mucking around with non-causal propagators did produce the requisite breakthrough.
It must be noted that Schwinger also had the same non-causal propagators, which he explicitly parametrized by the particle proper time. He arrived at it by a different path, from local fields. However they were both scooped by Stueckelberg, who was the true father of the modern methods, and who was neglected for no good reason. Stueckelberg was also working with local fields. It was only Feynman, following Wheeler, who derived this essentially from a pure S-matrix picture, and the equivalence of the result to local fields made him and many others sure that S-matrix and local fields are simply two complementary ways to describe relativistic quantum physics.
This is not true, as string theory shows. There are pure S-matrix theories that are not equivalent to local quantum fields. Feynman was skeptical of strings, because they were S-matrix, and he didn't like S-matrix, having been burned by it in this way.
I believe your puzzlement comes from confusing two frameworks: the quantum mechanical (photons) and the classical mechanics one, waves.
When one is calculating in terms of classical electromagnetic waves there are classical considerations : refraction, absorption, reflection with their corresponding constants .
When one is zooming in the microcosm and talking of photons, a wave is composed of zillions of photons which go through, each at the velocity of light.
The bulk of the target material is in effect the electric and magnetic fields holding the atoms together to form it: the nuclei are tiny targets and the electrons are small zooming targets. The probability of a single photon to scatter on a nucleus or an electron is miniscule. It interacts/scatters out of its optical ray path with the electric/magnetic fields that are holding the glass or crystal together. The scattering angles are very small in transparent materials thus preserving the optical path, enormous in opaque ones . It is those fields that one has to worry about, not the individual atoms and their excitations.
The photons scatter mostly elastically with the fields holding the solids together with tiny or high cross sections depending on the frequency of light and spacing of the materials. In crystals and glasses the optical frequencies have small probability of interaction.
x rays find most materials transparent because the photons' energy is much larger than the energies available by the fields holding the crystals together, and the scattering angles with the fields are very small, except when they hit the atoms, and then we get x ray crystallography.
Edit after comment
Here is the sequence as I see it:
A classical electromagnetic wave is made up of photons in phase according to the wave description.
There is an enormous number of photons in the wave per second making it up. Here is a useful article which explains how a classical wave is built up from a quantum substrate.
Each photon does not change the atomic or crystalline energy levels going through a transparent material in the quantum mechanical way by emitting a softer photon. It scatters quantum mechanically elastically, through the medium, changing the direction infinitesimally so that it keeps the quantum mechanical phase with its companions and displays transparency. Since a medium has a composite collective electric and magnetic field it is not a simple "electron photon going to electron photon" QED diagram. In the case of a crystal one could have a model of "photon crystal photon crystal" scattering amplitude for example.
The higher the sequential probability of scattering going through a medium the larger the final deflection through it will be, and the higher the over all probability of losing the phase with its companions in the wave.( the thicker the glass the less transparency and image coherence).
The transparency of the medium depends on the ordering of the atoms and molecules composing it so that it allows to keep the coherence between individual photons of the beam. The lower the density the better chance to keep the transparency, viz water and air.
hope this helps conceptually.
Best Answer
It is an analogy made for the wide audience. The actual picture provided by QFT is far more technical.
Configurations of electrons and positrons are states of the quantum system, called the electron field. This states can be obtained from the vacuum state by acting with creation operators.
This creation operators evolve in time (in Heisenberg picture), and the their frequency is positive for electrons and negative for positrons (or vice versa, it doesn't matter).
The whole thing is even more complicated in QED, when the interaction affects properties like mass, charge or field normalization factor via the renormalization.