I suspect that the passage is intended to say that they took the battery out of their phone, and then put both the battery and the phone in the fridge.
Assuming that's right, most likely there are two threats this is defending against:
Taking the battery out of the phone ensures that the phone is powered off. This defends against a malicious application or malicious code in the phone's firmware. (The concern is that someone might have attacked the software running on the phone and then installed malicious software or a malicious application that runs in the background and uses the microphone to record conversations. Taking the battery out of the phone powers off the phone's processor and prevents any such malicious code from running.)
Putting the items in the fridge provides soundproofing. This defends against a bug that was surreptitiously added to the phone. (The concern here is that an attacker might have, at some point in the past, planted a bug inside their phone. Such a bug might have its own battery and its own circuitry and its own microphone, and be recording their conversations. Taking the battery out of the phone will not stop such a bug. However, putting the components into the fridge will provide sound isolation, because the fridge is relatively soundproofed and also because the fan in the fridge provides a bit of additional noise.)
In other words, the first is about malicious software; the second is about malicious hardware. If you were only concerned about malicious software, it would be enough to take the battery out of the phone without putting it into the fridge -- but putting both of them into the fridge covers both bases. In their setting, this kind of paranoia makes sense.
Note that it wouldn't be enough to just put the battery into the fridge, but not the phone. The problem with that is that it wouldn't stop a bug that was surreptitiously planted inside their phone. That's why I suspect the quote meant to say that they took the battery out of their phone and then put both the battery and the phone into their fridge. As it stands, the wording of the quote is a bit ambiguous, so I realize I'm making a particular interpretation of the quote -- but given the two kinds of threats, I think my interpretation makes sense.