A parishioner showed me a beautiful devotional by Oswald Chambers that has a thoughtful statement in it about prayer. I believe it, and it reflects some of my lifetime journey of spiritual formation. Prayer is a gift of God, not something we "do" for God, or to try to get God to "do" something. Prayer is a gift of communion with divine love that can transform the human heart, perfecting us in love and maturity of faith. The traditional theological word for that is "sanctification."
I've always been bothered by the phrase "prayer changes things," not because I don't believe in prayer, or because I don't believe God can do anything God desires to do. On the contrary, to me the implication of the phrase limits the importance of prayer to a way we can get God to do the things we want. Of course, God can do the miraculous, and I think intercessory prayer (praying for others) is both important and lifegiving. Moses is a wonderful example of the power of intercession. It's not an incantation, it's conversation with God that can sometimes leave a glow on your face.
Chambers says it well here.
It is not so true that "prayer changes things" as that prayer changes me and then I change things; consequently we must not ask God to do what He has created us to do. For instance, Jesus Christ is not a social reformer; He came to alter us first, and if there is any social reform to be done on earth, we must do it. God has so constituted things that prayer on the basis of Redemption alters the way a man looks at things. Prayer is not a question of altering things externally, but of working wonders in a man's disposition. When yo prayer, things remain the same, but you begin to be different.