Why is Closure a necessary axiom in Group Theory? How can we explain the logic behind the closure axiom in Group Theory to a non-math major? What is the reasoning for why it is necessary? If the result of compounding two elements falls outside the group, why does that matter?

There is no logic to it. It's just how we define groups:

A magma just obeys closure, and is the simplest possible algebraic structure. Otherwise, you would basically just have a set (since the operation is meaningless). If I take the set $\{1,2,3\}$ and apply addition, there is nothing inherently special about addition to this set. I could just as easily apply multiplication. Closure is just a simple relation to link the set to the operation that closes it.
As for why groups in particular obey closure, it's simply because they are defined as a magma with particular properties (and any other combination just has a similar label). i.e groups are nothing special in terms of requiring closure, it's just a name we associate for structures that have the property of closure among other requirements
On an additional note, it's worth pointing out that the entire concept of algebraic structures is to generalise existing concepts. i.e since there are groups of cycles or matrices, if I prove something in general about a group, it will be true for both cycles and matrices. Closure is a common thing that most useful algebra (such as matrices and cycles) obey. Hence it would make sense for it to be a property of groups (so we can prove more lemmas, theorems etc).
