When the induction motor starts the rotor winding is stationary and maximum voltage is induced in the rotor which will produce maximum rotor current. At starting the rotor is stationary and acts in a similar to a transformer with its secondary winding short circuited except that there is an air gap in the motor. At starting the motor current can be like 6 times of the rated current. The voltage is reduced during motor starting to decrease the current but still the current is much higher than the rated current of the motor. When we say the current is reduced it is compared with 6x rated current.
When the motor is running at no load the speed is very close to synchronous speed. This is similar to a transformer with the secondary winding open circuited. When the motor is loaded the rotor speed drops and induces more voltage in the rotor that will produce more rotor current to increase the torque produced to counter act the mechanical load torque.
When the voltage drops the rotor speed will decrease because the electrical torque produced is proportional to the square of the voltage. As the speed decreases the rotor current will increase until it produces enough electrical torque that will counter act the mechanical torque. As the speed decreases the current becomes more reactive as a result the magnitude of current increases to produce the same active current (component of current that produces torque or shaft power). Induction motor is dynamic load and we can assume it as a variable impedance load.