Playtime,
I have wondered about this for years. The best guess I have is that it means something similar to “tzelem elokim.” It means that we have independent consciousness (a.k.a the mind, a.k.a. the soul) and ability to make moral choices. That independent consciousness, in contrast to our brains and bodies, are not the product of nature and physical processes but comes directly from on high.
In other words it doesn’t mean a piece of G-d, but a piece that is like G-d. It is not uncommon for Tannaic writing to use somewhat exaggerated writing (to leave out a “like”) or to use metaphor to describe something.