Aristotle's teacher was Plato. Plato taught that the soul is what causes movement. In simplified form, his argument is that only living things can move themselves, and all living things have souls, therefore, the soul is what causes things to move.
Aristotle thought Plato's thoughts on this subject were mostly garbage. Aristotle particularly disliked Plato's conclusion that the universe moves in an orderly fashion, therefore, it must have a soul. Aristotle outright rejected the idea of a "universe soul" that causes the universe to move, or a god that physically moves the universe.
This leads to the question of how movement works. Is it all physical? Humans can certainly be moved to action by non-physical things, such as ideas, the soul, or the subconscious. How do we account for those movements? Is it "cheating" to say "oh, those are just random chemical reactions in the brain"?
The problem we run into is the fact that all movement can't be physical. That just leads to a huge chain of movement, going from one thing to another. The chain can't go on forever, so it has to stop or start somewhere. Logically speaking, it can't stop with something physical, because that would just be another link on the chain; therefore, it must stop with something non-physical.
And that's why Aristotle claims there must be a non-physical thing, which affects the physical world. There is no other way to explain the existence of movement. He rejects the idea of a god like Atlas, who physically moves the universe, because such a god would, in turn, be affected by the world. As everyone points out, it's illogical that Atlas can stand on the world, at the same time he holds the world two meters away from his feet.