I took a sociology course in college and one of the main topics was spanking and how it is harmful. The professor taught us many reasons, such as the adult is showing the child that physical harm is an acceptable means of control, that the child cannot trust the adult, that since spanking is rarely a consistent behavior (especially because the parent is not around the child most of the time) this does not consistently reinforce the change of behavior, and the child learns that the appropriate first response to poor behavior is anger and causing physical pain.
From Research on Spanking: It’s Bad for ALL Kids:
It undermines trust. Children trust their parents just a little less. They are more likely to step back from the relationship and build a self-protective shield around themselves in terms of relationships generally. Children can learn to mistrust the motives of others and become more threat reactive in social situations. It can lead to aggressive expectations—they are ready to aggress first before they are aggressed against.
Spanking does not convey positive guidance on how to behave in a particular situation, only how not to behave if a threat of punishment is at hand. Children learn positive behaviors from practicing actions that work, ones that lead to a sense of belonging and competence. They internalize what they practice and what their family practices. They learn reasons for their actions from what they hear and are told, but active practice has the deepest impact.
Also see 10 Reasons Not to Hit Your Child from Dr. Sears, a well respected pediatrician.
You can put your reasoning is layman terms fairly easily once you learn the basic concepts. Try to think about what kind of explanation you would find helpful if you were being told the reasons. Think more about the most obvious reasons first, like hitting shows the child that hitting is ok and that it is traumatic to be physically punished. Then try to make the more scientific concepts simplified, such as how it has been demonstrated that this method of behavioral modification is ineffective, by saying something like how researchers have examined the problems with spanking and have proved that spanking over time doesn't not make things better. You can direct people to particular sites that have simpler explanations, rather than showing them direct studies. So essentially my suggestion is that you do extensive research yourself and once you have a fuller understanding of the many reasons why spanking is a bad choice then you will be able to easily take that knowledge and break it down for people who need it simplified. Hope that helps.