It's because of Faraday's induction law. A current through a coil will induce a magnetic field. The magnetic field will attempt to resist the initial flow of current (due to conservation of energy). This causes the back EMF.

To add to PerpetualAFK's answer, the geometry of a coil lends itself to fields crossing each other.

The back-emf is the same as counter emf (cemf). It is a voltage produced in a conductor that tends to neutralize the present voltage. It is a phenomenon that always tends towards the contrary of what happens!! Let´s suppose: if lines of a magnetic field are cut by a conductor, than a voltage is generated in this conductor, which causes a current of electrons to flow in one direction. At the same time, like trying to avoid this, ANOTHER voltage is created that tends to neutralize this effect, and it forces the electrons to flow in the contrary direction. As you know, the back-emf is directly pro- portional to the velocity of the magnetic field. It is proportional to the relative motion between them. The back EMF is the voltage that appears across the inductive reactance of the motor. It is still the voltage across the motor, so current flows into the motor based on the voltage of the back EMF divided by the motor impedance. When you say that the back EMF equals the applied voltage, it's really just saying that the voltage you applied to the motor can actually actually be measured there. If you applied a voltage across a resistor, you could also say that the current times the resistance is the resistor voltage and the resistor voltage equals the applied voltage. The back EMF is just one of the ways to be able to calculate motor currnets and voltages.

The counter EMF is due to energy being stored as the magnetic field of the coil. You don't actually need an AC current to create that field either. With AC, it will show up as a reactive impedance. In the case of AC being applied across the coil, as the AC changes directions, the coil will alternatley charge it's field, and collapse it's filed, driving reactive power into the rest of the circuit. That's why current lags in an inductive circuit.