Why does a changing magnetic field produce a current? A changing magnetic field induces a current in a conductor. For example, if we move a bar magnet near a conductor loop, a current gets induced in it.

Faraday's law states that 

The E.M.F. $\mathcal{E}$ induced in a conducting loop is equal to the rate at which flux $\phi$ through the loop changes with time.

Along with Lenz's law, 
$$\mathcal{E} = -\frac{d\phi}{dt}$$ 
Why is this so? The velocity of the electrons w.r.t. me, the observer is 0, so according to $\vec{F} = q\vec{v} × \vec{B}$, force should be zero in any direction on the electrons in the loop. Then what causes the current to flow and the E.M.F. to be induced? Is the force due to an electric field(the electric field in my reply to Albert in comments) or should I consider the velocity w.r.t. the source of the magnetic field?
Edit : I'm in a frame of reference which is stationary w.r.t. the loop, so from where does the electric force(due to the electric field in my reply to Albert in comments) come from?
 A: The third of Maxwell's equations (Faraday's law) says that a changing magnetic field has an E-field curling around it. The closed line integral of this electric field is the EMF that drives the induced current in the conducting wire. At a microscopic level, the curling electric field, which has a significant component parallel to the wire, exerts a force on the charges in the conductor.
If your question is "why are Maxwell's equations the way they are?", I'm afraid that isn't a good question for this site.
A: I guess you probably already know this, but still let me state it. If you have read Griffith's Electrodynamics or any other book on electromagnetism, one statement is always specified in faraday's law chapter...in different words

NATURE ABHORS A CHANGE IN FLUX
Note that it is the CHANGE  in flux, not the flux itself, which the nature dislikes. So according to lenz's law, it(nature...wire) will try to do anything to resist that change in flux. In your case, as the magnetic field increases when the magnet is brought closer to the loop, a clockwise current will flow in the loop as seen from the ammeter's side. This current is due to the electric field set up due to the change in magnetic field. It is given by the closed loop integral of $E.dl$ = -$\frac{\delta B}{\delta t}$A...since the area is unchanging. The electric field is circumferential which moves the charges in the wire....

If the charges are stuck to the wire......guess what happens??.....the wire rotates....which also means current(rotating charges, although stuck)!!!

Take a look at this,... The big , really long, cylinder is the magnet. As it is brought closer, there is an increase in the EMF. but it decays. Again as the magnet's end comes closer to the loop, the EMF spikes in the other direction

A: It is indeed a relativistic effect.
In fact you can derive Faraday's law from the Lorentz transform of the electromagnetic field.

A boost (velocity) orthogonal to a magnetic field $\textbf{B}$ transforms in an electric field $\textbf{E}$ that is both orthogonal to the boost (velocity) and the $\textbf{B}$ field.
Only the component of $\textbf{B}$ in the plane of the ring is involved because this component $\textbf{B}_\perp$ is orthogonal to the boost. See the green highlighted term in the Lorentz transform of the electro-magnetic field. The term




*

*$\textbf{B}_\parallel$ = component of $\textbf{B}$ parallel to the boost.

*$\textbf{B}_\perp$ = component of $\textbf{B}$ orthogonal to the boost.

*$\textbf{B}_\bigotimes$ = same component as $\textbf{B}_\perp$ but 90 degrees rotated with respect to the boost direction.

*At low speeds $\gamma \approx 1$ and $\beta$ is proportional to the velocity.


Now note that only $\textbf{B}_\perp$ contributes to a change in the total flux through the ring while moving the magnet, and that the term $\textbf{B}_\parallel$ does not contribute.
This is how you can derive Faraday's law from the Lorentz transform of the electro-magnetic field.

http://www.physics-quest.org/Book_Chapter_EM_LorentzContr.pdf
A: As you stated right it has to be a conductor to induce a current from a changing magnetic field. In conductors there are some electrons which are in unbounded states to the nuclei.
It is well known that a magnetic field and an electrons electric field don't interact. So what is interacting? The electron has two more properties, one of them is the electrons magnetic dipole moment.
If a stationary electron is inside a constant magnetic field the electrons magnetic dipole moment gets aligned with the external field and that was is, no current will flow.
If you are "riding" on a changing magnetic field you will see the electron (in the wire) moving (in relation to the changing magnetic field). And there is a phenomenon of deflection of moving charges, called Lorentz force. If you are sitting on the electron you get the same picture, regarding a changing magnetic field the electron seems to move and by this gets deflected. So in both frames the relatively movement between charge and field (no matter does from a third frame the charge moves or the magnetic field), there will be induced a current.
Now the clou. There is a phenomenon of a homopolar generator where both the magnetic field and the disc are rotating together and nevertheless a current is induced:

A: When  the  north Pole of magnet is approaching towards the coil ,the coil  facing the north Pole  behave like a north Pole and both of tries to repel each other and they will try to stop the motion of magnet, so to push the magnet​ against repulsive force work must be done against repulsive force , the work appears in the form​ of electrical energy in anti clockwise  directions. ......,......Read this it is complete explaination
