HOW is an electric field induced when there is a time varying magnetic field? I have just learnt electromagnetic induction. When the magnetic flux through a closed loop changes with time, a current is produced in the loop. My teacher told me that this was due to an "induced electric field" in the wires, and that it was a non conservative field. 
How did this field magically appear? I would like to know the mechanism- or at least an intuitive understanding of how this happens. I am also aware that a current carrying wire exerts a magnetic force on a moving charge due to length contraction, responsible for creating an extra electrostatic force on the charge. Can relativity be used to explain Faraday's law as well? Even Griffiths does not get into what causes the induced electric field.
 A: The field didn't appear "magically". Electric and magnetic fields are part of one and the same phenomenon: electromagnetism. They can not be separated. The only reason why we teach electric fields and magnetic fields independently, at first, is to make them easier to understand for students. 
When you are learning electromagnetic induction, you are being exposed for the first time to the fact that what you falsely thought were different phenomena, are just two corner cases of one and the same phenomenon. That phenomenon is described in its entirety (at least on the classical level) by Maxwell's equations, which treat electric and magnetic fields in a unified way. 
So instead of thinking about an induced electric field as "magic", you should learn to think about it as natural. What is unnatural is that we don't teach it in a unified way from the beginning. That, on the other hand, is just a side effect of over a century of educational history, and one could argue that it's probably time to stop telling "lies for children" about electromagnetism.
Instead we should start the educational units about electricity by saying that there is one electromagnetic field that has an electric and a magnetic component and that these components are linked very intimately. Only when neither field changes in time, can they be though of as decoupled (and even that is not completely correct, but it would suffice for most, if not all, demonstrations that one can do in school). 
A: 
How did this field magically appear?

Induced electric field is caused by the variation of current-density in the stationary loop and not by magic. The electric and magnetic fields are correlated by 
$$\mathbf \nabla \times \mathbf E~=~ -\partial_t\mathbf B\;.$$

I am also aware that a current carrying wire exerts a magnetic force on a moving charge due to length contraction, responsible for creating an extra electrostatic force on the charge. Can relativity be used to explain Faraday's law as well?

I could conclude that you were saying A magnetic field is caused by length contraction and....
Well, 
Magnetic field is due to the relativistic effect of electric field.
But in the same way,
Electric field is due to the relativistic effect of magnetic field.
That is, taking one field as the mean the charges interact, the relativistic transformation equations make the presence of the other field imperative.
But, both electric and magnetic field can't be relativistic effect at once.
As Jefimenko in his paper points out:

The only correct interpretation must be that interactions between electric charges that are either entirely velocity independent or entirely velocity dependent is incompatible with the relativity theory.Both fields—the electric field (producing a force independent of the velocity of the charge experiencing the force) and the magnetic field (producing a force
  dependent on the velocity of the charge experiencing
  the force)—are necessary to make interactions between
  electric charges relativistically correct. By inference
  then, any force field compatible with the relativity theory
  must have an electric-like ‘subfield’ and a magnetic-like
  ‘subfield’.

The electric and magnetic fields are the two aspects of the electromagnetic field. It depends on frame whether electromagnetic field would like an electric field or magnetic field or combination of both.
See Christoph's answer here.
Also, see the last part of Timaeus' answer here.
