There have been studies on the subject, having people walk or not (and walk in varying conditions) and measuring their performance on some intellectual or creative task, and concluding that (a) walking does help and (b) the type of walk probably matters.  First citation I found: https://www.apa.org/pubs/journals/releases/xlm-a0036577.pdf

Four experiments demonstrate that walking boosts creative ideation in real time and shortly after. In Experiment 1, while seated and then when walking on a treadmill, adults completed Guilford’s alternate uses (GAU) test of creative divergent thinking and the compound remote associates (CRA) test of convergent thinking. Walking increased 81% of participants’ creativity on the GAU, but only increased 23% of participants’ scores for the CRA. In Experiment 2, participants completed the GAU when seated and then walking, when walking and then seated, or when seated twice. Again, walking led to higher GAU scores. Moreover, when seated after walking, participants exhibited a residual creative boost. Experiment 3 generalized the prior effects to outdoor walking. Experiment 4 tested the effect of walking on creative analogy generation. Participants sat inside, walked on a treadmill inside, walked outside, or were rolled outside in a wheelchair. Walking outside produced the most novel and highest quality analogies. The effects of outdoor stimulation and walking were separable. Walking opens up the free flow of ideas, and it is a simple and robust solution to the goals of increasing creativity and increasing physical activity.

Another hypothesis: walking turns on your brain's "we're going somewhere, let's keep track of where we've been / we are / we're going" mode. But since you're not going anywhere, and your path is very simple / doesn't require any high-level/deliberative adjustment or attention, that "journey mode" energy is purposed towards your thinking journey.

Slightly related: people tend not to like it when I pace in their presence for more than a minute. My hypothesis is that it feels to them like I am asserting dominance over the room (more so than if I were to dominate the conversation, less so than if I were to issue a string of commands).

Skilled speakers often pace or wander among the audience, which is evidence for my hypothesis because audiences tend to want the speaker to dominate the room when they have come to the room to hear a speech.

When going for a walk, you are somewhat far from your desk, but if you're pacing somewhere around your house, your desk is nearby. This means that it is quite low friction to switch between working and pacing.

I have been walking back and forth a 50m trail for hours as part of a meditation exercise. It was quite a deliberate effort. I concentrated on the movement of my hip bone and learned a lot from it. But I can't say that it is "inexplicably good."

Pacing is a common stimming behavior. Stimming is associated with autism / sensory processing disorder, but neurotypical people do it too.

When I look for a place to stay, I actually specifically try to find a place where I have enough room to pace around the house/apartment in a somewhat straight line back and forth.

I also really prefer offices that have these sections where I can pace around while thinking (especially when it's isolated from others so I don't distract them). I think anyone in EA who is looking to purchase or design offices should keep this in mind.