Since it wasn't exactly spelled out, you could only use a Y-Delta starter with the origional motor in question if the supply is 240V.
You start in Y mode with the motor expecting 415V. This means the 240V souce is only 58% of the motor rated voltage. This effective reduction in supply voltage lowers the motor current and torque to provide a softer start.
Then, you switch the motor to a delta connection after some period of acceleration time. At this point, the motor is connected properly for the 240V supply and you will be able to get rated torque out of the motor.
You can not use a Y-Delta starter on a motor when your source is the higher of the two rated motor voltages. In this example, you can not use a Y-Delta starter on a motor that is 240V/415V rated if your source is 415V.
If you are using a 415V motor with a Y-Delta starter then the motor could theoretically be a 415V/720V rated motor. However, it would likely only be listed as a 415V motor on the rating plate with the 6 leads being supplied to allow the use of a Y-Delta starter, not to allow a dual voltage rating.
If you don't know, the conversion factor between Y and delta voltages is the square root of 3.