Why Does Change of Magnetic Flux Induce an emf? Why does change in magnetic flux with time through a coil induce an emf across it? Please explain what happens to the charges in the coil when magnetic flux changes?
Also, why does a constant magnetic flux not induce an emf?
 A: If you move a wire through a magnetic field so that the wire, the motion, and the magnetic field are all mutually orthogonal, them there is a magnetic force on the charges (because of the motion) that has a component in the direction of the wire (because we arranged the directions the right way).  The energy comes from the agent moving the wire, but there is a real force.
When electric and magnetic forces were considered separate, it was merely an experimentally observed fact that if you changed the magnetic field in time there would be a force driving the charges through a wire.  But if you look at that magnetic force above, but consider the situation in the frame of the moving wire, there is no magnetic force in that frame, but there is still a force.
You can consider that to be the real reason that changing magnetic fields induce electric fields.  
A: My answer is not associated with the forces on electron and changing electric field.I am giving this answer which is just to feel what happens.
Change is the law of nature but, no one likes sudden changes. U may call it analogous to the first law of newton relating to inertias. In the similar way, suppose a loop is kept in a magnetic field with a magnitude of B tesla. B may be equal to zero. We also use crosses and dots in a plane to show magnetic flux density. Suppose a total of 5 crosses representing B tesla,  is passing through the loop. Now, if we


*

*Increase magnetic field say, as a function of timeThe loop had learned to live with 5 crosses (x), but now it has say 6 (x). It must have to  find a way to get back in the position of 5 (x)and to do that it induces current in such a direction which produces dots (•) inside the loop, watching from outside the computer screen, the current should be anticlockwise (according to screw rule).

*Magnetic field decreasesIn the second scenario, suppose crosses decrease to 3. Then loop will induce a clockwise current to bring in some crosses to get back to the position of 5 x.

*Area of the loop changesYes, this can also change the flux and inturn induce current. Flux is defined as 
Magnetic flux = $(Magnetic field)(Area)\cos\theta$ where $\theta$ is the angle between area and field vector.
And emf induced is the derivative of Magnetic flux. So if Area changes - flux changes and thus emf has a non zero value.
Again this could be easily understood by my above theory as again the number of dots and crosses changes giving rise to current


*neither magnetic field changes neither area
Or magnetic flux remains constant.
Now its obvious that if u are in the same position as before without any change then why would u like to do anything else. I hope u understand what i am trying to say


Hope it helps u and reasons based 
on the forces and electric fields are explained already in others' answer.
.
A: It is an experimental fact that a changing magnetic field induces an electric field and a changing electric field induces a magnetic field. This has been mathematically described in the all inclusive Maxwell equations. 
Conductivity appears with the motion of charge in a conductor.The difference between insulators and conductors is that in insulators the electrons around the atoms and molecules composing them are bound and cannot be detached from the potential that binds them to the nuclei of their atoms/molecules. In conductors some of the electrons are bound by a collective potential of the material, in bands where they can move large distances with respect to atomic distances, being shared with many atoms/molecules.
When an electric field is applied on a conductor the electrons are attracted to the poles and move in the direction of the electric field and a  current appears. In a closed circuit conductor a changing magnetic field will be producing an electric field to which the electrons will respond by moving in the direction pointed by the field. 
It is the changing magnetic field that induces the electric field. A steady one does nothing.
A: To produce EMF we need to drive electrons or we need to produce a net charge difference at the ends of conductor. As we all know that a moving charged particle will experience force by a magnetic field, so, if we are talking about dc machines, the magnetic field produced by the fieldcoil remains constant, that is, no force is experienced by electrons  of the wire. So, we have to produce a relative motion between magnetic field and electrons. Whenever the flux passing through the coil changes by any way (like either changing angle, magnetic field or area of coil), we are actually producing a relative motion  between electrons and magnetic field. As a result, the electrons experience  a magnetic force and shift to produce EMF.
