As a former math-head, let me chime in. The parent poster is probably not looking for an explanation of what exchange operators are, which are rather abstract and uninteresting in a physical vacuum, but rather how they are related to physical reality.

I’ve learned a lot about both and still am not sure why exchange operators and physical meaning are related. Why does the fermion care if you do an exchange operator to it? I get that fields of fermions demand that behavior but it still feels abstract.