How does an Inductor "store" energy? It seems to me that an electromagnetic field is nothing more than a collection of photons, which as I've heard, extends through space infinitely. Why is it, then, that an inductor such as simple copper wire loop, can "store" energy in it as an electromagnetic field? Wouldn't the photons or waves of EMF just fly away into space and be lost (the energy would be lost, not stored), how is it that this energy is stored as if the photons would fall back down and hit the wire to create current when the field collapses?
 A: Your argument that the energy should radiate away would be true if your inductor were a good antenna, in which case it would be a bad inductor! The problem is an impedance mismatch: The inductor produces a magnetic field (which stores the energy you inquire about), but little electric field. That is the wrong ratio, or impedance, to couple to the vacuum where photons travel at the speed of light.
You obviously are correct in arguing that this is nevertheless electromagnetic energy that must be quantized as photons. But these photons are localized, essentially trapped inside or in the neighborhood of the inductor.
A: 
Wouldn't the photons or waves of EMF just fly away into space and be
  lost (the energy would be lost, not stored)

One must distinguish between electromagnetic waves and, e.g., static electric and magnetic fields.
Essentially, ('real') photons are associated with electromagnetic radiation (radio, light, x-rays, gamma rays).  Electromagnetic waves transport energy and momentum far away from where they are generated.  Quantizing electromagnetism results in quanta, photons, that have both energy and momentum.
But static or (relatively) slowly varying electric and magnetic fields are not electromagnetic radiation.  A static electric and / or magnetic field does not transport energy but we can associate an energy due to the configuration of charges and / or currents.
In the case of an inductor, work is done to establish the magnetic field (due to the current through the inductor) and the energy is stored there, not delivered to electromagnetic radiation ('real' photons which would indeed transport the energy and momentum elsewhere).
Now, you may wonder how such static fields are treated quantum mechanically and this necessarily involves the notion of 'virtual' photons which is beyond the scope of this answer.
A: A moved charge - in this case an electron - which is accelerated in a circle (the inductive coil) will induce a magnetic field. How does the electron induce the magnetic field? The electron has a magnetic moment and it spins. The movement of the electron in the coil align the magnetic moment and all moved electrons induce the common magnetic field of the coil. By this the electrons get slower, in macroscopic terms the resistance of the coil increases. At the moment when the current will be interrupted the magnetic moments fall back and when starts the self induction, the magnetic field collapses and give back the energy to the electrons. 
More about the Lorentz force see How does the Lorentz force work?
