But there is something about that phrase that makes it very effective, even if your higher power is not God. It states your intention to stop trying to control things which you cannot control, and instead, do the things you do, play your part, but trust in the universe, your higher power, to either make it happen or not.
We want things to happen a certain way, both regarding the thing and how it comes about. By "letting God," we are not only giving up our desire for control but we are giving up the nature of the actual end result. The universe may feel that what we want is not in our best interests, at least not now, and so it doesn't happen despite all our efforts.
The point is by "letting God" we acknowledge that, as the saying goes, "the Lord acts in mysterious ways." The universe will know what is best for us and that is what will transpire. If may be, and mostly likely will be, very different from what we have in mind, what we desire, but we have faith that our higher power is here to protect us.
By letting go and letting God, we find peace because we have let go of the craving and attachment associated with our desire and so that weight is lifted from our shoulders and we feel free.