OKAY. Trying not to shout. Really. Trying. You rein in spending or anything you are trying to slow down or cut. You don't reign it in. Why? Both are metaphors. The first, rein, is what you use to control a horse. Oh my. Oh my. I am so old. I have actually, in my lifetime, ridden a horse. That's how I know.When you pull back on the reins, you are pulling the horse's mouth closer to its neck, which makes it difficult, if not impossible, for him to run. (Conversely, when you " give it it's head" -- referring again to a horse, you loosen your grip on the reins, which allows the horse to stretch out its head and allows it to run if it wants to.)
Second. People who do what they are told, toe the line. They don't tow the line. Again, we are talking metaphor. You know what actors do when the director yells "places"? There are little lines painted on the stage. The actors put their toes on the line for their place. They toe the line. I don't know where this "tow the line" comes from. Pulling a rope behind you? I don't know. If you have any other important questions please let me know. It's my inner English teacher.