Well, if I can add my 2 cents' worth... This is the way I understand it:
First off, EMF is not voltage. EMF is an electromotive force (collective Homer-like "duhs" are NOT appreciated.
). If there is a path, EMF is current. If there is no path, EMF is voltage. Normally there is a partial path which we call a load and EMF becomes a combination of Volts and Amperes, which we call "Voltamperes" or VA.
Now, since most people relate to EMF as it applies to generators, and generators have voltage regulators, the EMF produced is typically thought of as a fixed voltage with a varying current, giving us the maximum current part of its rating, which is really in KVA. One states that you can measure EMF in a motor by measuring the voltage. All of this is correct, but I hope my first paragraph of explanation helps clarify, somewhat, the concept of EMF.
Now, due to Farraday, we know that we get an EMF in a conductor when it has relative motion to a magnetic field. We call this type of EMF production: induction or induced EMF.
The stator has an EMF applied to it from the line. As the stator is a conductor and a complete path exists, a current flows and produces a magnetic field. This magnetic field grows and shrinks and changes polarity with the applied EMF's changes in amplitude and polarity.
This moving magnetic field cuts the stationary rotor conductors (squirrel cage or wound rotor work the same). The relative motion between the rotor windings/bars and the moving magnetic field produces an EMF in the rotor through induction. As the rotor is shorted with the end rings (assuming a squirrel cage), the EMF becomes a current. (Note: motors are made with superconductors for this explanation; the voltage component caused by IR drop is ignored.)
Now the stator has a current-produced magnetic field and the rotor has a current-produced magnetic field. At this instant, we have what is essentially a shorted transformer with its associated infinity-seeking current. The current is limited by the inherent resistance of the stator windings, the saturation point of the laminations, and the resistance of the rotor, and of course the impedances of the source. This high current is what we term "inrush current." Also at this instant, since we have not yet gotten rotation, all we're doing is making magnets in both the stator and rotor.
The rotor's magnetic field is produced by induction from the stator's excitation. By Lenz's law, ("An induced electromotive force generates a current that induces a counter magnetic field that opposes the magnetic field generating the current.") the magnetic fields must be in opposition to each other; producing a repelling force between the rotor and stator. We call this force "torque." Torque is essentially the strength of the rotor magnetic field times the strength of the stator magnetic field.
As an aside, I disagree with the first reference cited in that I understand motors to be turned due to the opposition of magnetic fields, not the attraction of them.
At this point, the fields are pretty much saturating the cores due to the high currents in both the stator and rotor, so maximum torque is produced; no more flux coupling is happening because we've run out of iron. This accounts for the changing torque on the torque to speed curve as the rotor changes its rotational speed and the inherent magnetic induction changes the rotor's EMF's frequency. In other words, as the rotor changes speed, the frequency in the rotor becomes less because the relative motion between the fixed applied EMF/magnetics and the moving rotor is decreasing. Lower frequency causes induction to become less efficient. Likewise, as we'll see shortly, stator current will also be decreasing causing a related lowering of magnetic field strength, but this is just detail illustrated by the curve.
Now, the rotor must be magnetized to get torque, and that magnet is now moving as it was meant to do. This produces an EMF in the Stator due to Farraday, once again (moving flux, stationary windings = relative motion). Now, current can't flow in two directions at the same time, so the induced and opposing EMF in the stator that is produced by the rotor cannot overcome the applied EMF from the line, as the applied EMF is the source of energy for this whole process. We call this opposing EMF Counter EMF or Back EMF.
Therefore, this induced EMF is presented as a voltage on the stator (remember, it can't be a current because that would be two currents flowing in opposite directions simultaneously) which can then be subtracted from the line-applied EMF/voltage on the stator, causing the apparent voltage (terminal voltage) felt by the windings to decrease. Therefore, the stator winding current decreases with this decrease in its perceived voltage.
Likewise, the rotor is coming up to speed and its current is decreasing as well due to the decreasing frequency difference between the applied stator flux and the rotational speed of the rotor. If the rotor's speed were to equal the applied flux's frequency, then there would be no relative motion between the windings and the flux, and no EMF would be produced in the rotor.
This loss of EMF causes the rotor to slow down due to loss of torque, and that causes the relative motion to return. This is a dynamically established balance within the motor; the rotor will turn at the proper speed to produce EMF required to produce the torque required to turn the motor at that speed.
When the load is increased on the motor shaft, the rotor slows, causing a greater difference between the rotor's speed and the stator's flux, causing more EMF in the rotor and therefore more torque. However, since the rotor is going slower, it produces less EMF and the stator's applied voltage therefore has less opposition and therefore its current goes up also contributing to an increase in torque.
I hope this wasn't too wordy. I tried to explain this in simple terms to the best of my understanding. I welcome criticism as I've arrived at this through my own meanderings, not from reading it in a text. The texts always seem to rely more on obfuscation and formulae rather than simple explanations of what's really happening.
I hope it helps.
Cheers.