Why 'emf' is induced at the ends of coil when a bar magnet moves inside the coil or vice-versa?

It is because the electrons inside that coiled wire experience a force due to the CHANGING magnetic field (changing with respect to space coordinates). When the magnet moves to one side , the electrons experience a force in a particular direction. When the magnet moves in the opposite side , the electrons ,too, feel a force in the opposite direction(relative to their previous experienced forced).

Now WHY does this really happen, and i mean the 'actual' why , is not known clearly with absolute certainty.To understand the reason behind this 'why' , one has to research in time,space, nature and origin of light...it's interaction with matter (and that precise mechanism).Yes it involves all mathematics and quantum mechanics.Humans are still researching.