You're looking for the really thick tuberous bit called a rhizome. It'll be dark, woody, and have a tree bark texture. In the fall, when the lily starts to die back, is probably the easiest time to do this.
The rhizome for a lily like that will probably have numerous sections, or even have split all by itself. If you find it's one long rhizome, look at where the shoots have come out and when cutting, make sure each section has at least one shoot. You can rinse off the dirt with a garden hose to see better.
Another advantage to doing this in the fall is that you can cut away almost all of the roots without any problems.
