Reasoning is a logical, thoughtful way of thinking. When your teacher explains the reasoning behind his classroom rules, he makes it clear exactly why and how he came up with them.
Voters often want to understand the reasoning behind certain laws, and toddlers almost always want to know the reasoning behind rules about bedtime and wearing warm clothes on cold days. You can also use reasoning as an adjective, to describe someone who can think logically. A reasoning adult can make decisions for herself. The word comes from reason, which is rooted in the Old French raisoner, "discuss or argue," and the Late Latin rationare, "to discourse."