Some days ago, after a long conversation with a friend, I took retrospective notes, recalling what we discussed. (I wrote over 50 items. This is not a small sample.) For parts of the time, we walked; for other parts, we stayed in roughly the same place. Writing the notes, I could tell I was missing up to half of what we said (and forgot the order) for those stationary parts. I had no such issues recalling the walking-concurrent parts.
I've noticed this kind of discrepancy a bit before, but this was when it became obvious. Walking as you talk can help a lot towards remembering what's said.
This may be a quirk of how my mind works and wouldn't apply to everyone. I doubt I'm totally unique, so it probably at least applies to some other people, even if not everyone.
This part is justified speculation, not explicitly confirmed, and not important if you're just using the method.
What seems to be going on here is that my episodic memory encodes the surroundings I see together with the conversation I hear. Later, when I recall the event, I think of what I saw around me to help bring the event to mind, and what I heard (or interpreted at the time from what I heard) comes with it.
I see two ways that the walking helps:
I don't know which is more important.
The mechanism here depends on you moving, not those with whom you speak. If you only care about your own memory (or the circumstances otherwise demand it), you could call them while walking alone and get the same effect. (I have tried this. It works.)
The method should also apply equally well to one-sided speeches to which you listen, tho those tend to be more predictable anyway.
The mechanism here depends on movement and changing surroundings, not specifically walking. I expect you'd get the same effect if you're cycling/driving/riding a vehicle, so long as you're observing what you pass by as you do so. That happens to be easy in the case of walking: you have no vehicle to obstruct your vision, and you're exposed to mild, attention-demanding risk from every direction.
If you care about the order in which things were said, don't go over the same place twice. Repeating locations makes your path overlap with itself, complicating sequential recall. (This mistake caused a bit of confusion in my example at the beginning.)
Apparently, the brain models some kinds of abstract spaces similarly to how it navigates in real life. (Citation needed. Relevant keywords "hippocampus" and "have you ever played a modern video game?") You might get the same effect if you move in an intricate video game while you talk. That might require full VR. (I have not tried this.)