I'll have to leave the full "how" you want to someone else who can explain it better...
"why" in physics is sometimes a philosophical question.
Why is Vemf=N*d/dt(B) (Faraday's Law)? That's because nature works like that. ;)
How can we get to Faraday's Law from other concepts?
"How an magnetic field get converted into electric current ?"
Magnetic fields , are , by definition, the result of electrons moving with a velocity (which is current).
Hence, if you have a wire with current flowing through it, there is, by definition of a magnetic field, a magnetic field around the wire caused by the current flow.
Conversely, if you have an external magnetic field, and you place a wire in it, current can be produced. Remember, however, that magnetic fields only act on moving electrons (current). So, if there's current in the wire, a force will act on it in this case (Ampere's Law).
If the conductor moves, there's moving electrons, so the external magnetic field will induce an EMF on the conductor. However, if there's no conductor movement, there's no moving electrons, so the magnetic field won't effect the conductor (remember, magnetic fields are caused by and only influence MOVING electrons).
Note: this is a gross oversimplification. There is of course a natural drift of electrons within the conductor, but there's no net movement.
I hope this helps.