I apologize in advance if this isn’t the right place to post this, but I though some of you might know the answer to my question.
Welcome. This question is arguably off-topic because it is about the cell network which is not amateur radio, but it can be answered on general principles of radio communications, so I'll do that.
what I don’t understand, is how one’s phone can transmit RF signals while not in flight mode, but not in ringing/call mode either. What does the phone transmit to the cell tower when it’s not receiving/is in a call?
The cell network needs to know where the phone is.
Suppose that there were no such transmissions and you hadn't used your phone for a few days. Then when someone called you, every cell tower in your provider's coverage area would have to transmit that message and listen for your phone's response, because you could be anywhere as far as the network knows. This is a scaling problem: every call started would be occupying some of the network capacity everywhere.
Instead, your phone regularly transmits short messages, and whichever cell towers receive that message well are the ones that will transmit when an incoming call or other data needs to get to your phone. The same thing happens continuously while you are in a call to ensure that audio always gets through even if you move (unless you move to an area not covered by any cell).
Another factor is of course that phones today have many “apps” and other features which are not just making calls but transferring data for other reasons — these are other reasons to transmit that are not making a call. But those are really just the same as calling from a high-level perspective of how the network and protocol works.