Short Answer
Philosophy is almost completely conducted in natural language, and the study of the origins of the "why's" of natural language are a concern of the philosophy of language. What these philosophers agree on is that metaphors are used to communicate thoughts. In fact, some philosophers believe there is a language of thought. Ultimately, the use of "move" in the context you are talking about is drawing a comparison between having logical arguments and some sort of motion.
Long Answer
When communicating ideas, it often helps in philosophy to use our intuition about familiar things to talk about other things. There's no more famous advocate for this than Daniel Dennett who has a book entitled Intuition Pumps and Other Tools for Thinking. He has sections on philosophical razors like Occam's (no, it's not a real razor) and brains which compute (is the brain a computer?). In fact, when you have something 'in mind', do you really think your mind is an actual container with things called ideas in it?
You see, many people come to philosophy with the belief that formalisms like symbolic logic are some sort of decisive practice to separate truth from falsity. Nothing can be further from the truth. While symbolic logic is a fantastic tool for exploring ideas, the brain is far better constructed to use natural language in the context of informal logic. This means that most philosophy, while aided by equations and whatnot, actually relies heavily on intuition.
What does our intuition tell us about debate? It tells us it is a conflict. It tells us it is like a game where one opponent tries to defeat another opponent. It suggests that words fly back and forth (especially in a hostile debate) in the same manner flights of arrows might be loosed from two enemy lines at each where the job of the generals is to respond to changes on the battlefield and decide where to move their truths. This is why philosophers often use language like "attacking and defending a position". In the physical world, there is not a position. There is no physical attack that needs to be defended. But it helps us to understand what is happening when two philosophers are engaged in a back and forth. Sometimes, philosophers even get cheeky and exchange barbs. My favorite is Francis Crick's from 282 of his The Astonishing Hypothesis:
Dennett is a philosopher who knows some psychology and also a little about the brain. He has interesting ideas but appears to be overpersuaded by his own eloquence.
Can't you feel the adversarial nature of the philosophical discourse, like two ancient wizards in a duel summoning their best spells to try to show their position is the better position? This is what metaphor in language is all about.
In fact, the noted linguist George Lakoff turned philosopher has a thesis in his book Metaphors We Live By that states in essence that at a fundamental level of neural computation, all of our language is infused with metaphor. 'Up is Good' explains why 'feeling up and down' means 'feeling good and bad' or 'Love is Motion' explains why we express love as 'falling in love' or being 'swept off our feet'. Later, in his Philosophy in the Flesh, he and his coauthor give an argument about how this stems from neurocomputation of language. This notion is known to those of us who have an interest in the philosophy of language as a conceptual metaphor inline with cognitive semantics.
Other phrases you might see are "planks" in an argument, "horns" of a dilemma, and "levels" of understanding. In fact, if you start to scrutinize language, you'll find so much metaphor in language, you'll begin to detect what someone believes based on their surface structure. Does your lover "invest" themselves in you or do they "support" you? In one case, the relationship is viewed as a resource, and in the other, a structure, both of which are metaphor.
