Technically this isn't true, at least not as a blanket statement regarding all phones.
Since the power button is a 'soft' switch, controlled by the phones operating system, it's possible that a phone can be compromised so that even if it appears to be 'off' it is actually still powered up and listening to the cell network, but this is in no way a 'standard' feature. The same attack vector/vulnerability is present for flight mode, a compromised phone can lie and report that the radio is off when malicious software is using it.
Removing the battery is a mechanism to preclude any possibility of your phone being exploited, and may be appropriate depending on your risk tolerance and/or threat model.
Since the power button is a 'soft' switch, controlled by the phones operating system, it's possible that a phone can be compromised so that even if it appears to be 'off' it is actually still powered up and listening to the cell network, but this is in no way a 'standard' feature. The same attack vector/vulnerability is present for flight mode, a compromised phone can lie and report that the radio is off when malicious software is using it.
Removing the battery is a mechanism to preclude any possibility of your phone being exploited, and may be appropriate depending on your risk tolerance and/or threat model.