Plato simply asserted that there was a soul, however, if he was forced to show evidence for it, it might turn out something like this:
Do you believe that there exists something other than the physical?
We can simply observe the nature of thoughts and show that some do not originate in the physical.
For example, we impose catagories on physical objects that are artificial, and reasonable in nature. The number 2 is an example of this. We never see the number 2 in nature, simply things that fall under the catagory of 2.
If these non-physical thoughts do exist, where do they originate? Surely, the very fact that you are able to observe your own consciousness and physical body from another perspective denotes a soul.
Reason itself is the property of the soul and testifies to its existance
Anyway, thats just something I think Plato might say. Although he would probably fluf it up in a dialogue between Socrates and Crito or something.