In his Physics, Aristotle conceives of causes in several different ways. Our everyday notion of cause and effect (like the cause of the tree falling is that I cut it down) is only one of the ways in which Aristotle conceives of causes. In regards to the first cause of existence, or the unmoved mover, he is thinking of a type of causation which is much different from cause and effect. The unmoved mover is not the first pebble thrown into the pond of existence- it did not create a cause and effect chain which now constitutes the movements of this universe. Instead, the unmoved mover "caused" everything in the universe to have its purpose or its "for the sake of which" that assigns a purpose and identity to everything in the universe. For example, the "cause" or purpose of an acorn is to grow into a tree. The purpose of a tree is to be a tree which will grow leaves and produce more acorns. The purpose of a chair is to be sat upon, the purpose of a pen is to be something to write with and the purpose of rock is to be hard and motionless. The unmoved mover is responsible for creating the order that is now existent in the world- it assigns different things different roles and defines what these roles are. Without the unmoved mover, matter would have no form, no purpose and no motion.
While you may want to picture the unmoved mover as a type of God, it is safer to think of this entity as a first cause, a thing which created the universe, leaving out the anthropomorphized notions of God that we typically associate with divine beings.