Why is it impossible for induction motor to operate at synchronous speed?

Answer 1

Torque is produced because the rotating magnetic field, set up by the stator windings, cut and induce voltages into the rotor bars.  The fields set up by the resulting rotor-bar currents then interact with the rotating field, causing the rotor to turn.  If the rotor turns at the same speed as the rototing field, then its bars will not be cut by the field, there will be no induced voltages, no bar currents, and the torque will disappear.

Answer 2

The torque produced by an induction motor depends on the current induced in the rotor. To have current induced in the rotor there must be a difference between the speed of the rotor and the speed of the rotating magnetic field. At synchronous speed the rotor current is zero and the torque must also be zero, so it never happens.
An induction motor runs at less than synchronous speed by an amount called the 'slip', defined by a formula:
s = (Ns - N) / Ns
Ns is the synchronous speed and N is the actual speed, so with a slip of 0.1 or 10% the actual speed is 10% less than the synchronous speed.
In a simple model of an induction motor the current drawn and the output torque are both inversely proportional to s.