Meaning of slip in induction motors
However the only way I can imagine magnetic field and rotor having different speeds is if the angle between the two is constantly increasing. Is this what is actually happening?


Yes.

However, we normally express 'angle is constantly increasing' as 'angular speed is different'.

If the field is rotating at 50Hz, and the rotor is rotating at 47Hz, then the slip frequency is 3Hz, or 6%. This is the effective frequency seen by the 'transformer' whose secondary consists of the shorted rotor circuits. This is also a convenient explanation of why the torque of an induction motor would be zero at 0% slip, because the transformer will not induce any current at all at 0Hz.