They are called proliferations, which is actually a new daylily plant. Leave them on until you see air roots develop, the cut them off, leaving part of the scape and pot it up, or you can place them in a small container with water up to the roots and allow the roots to grow a little more, they will be exactly like the parent plant. Here is a link explaining it better:
Thank you cajunbelle! :-D I was really puzzled by this. I'd never seen it before. I'm glad I can make new plants from it. These were beautiful. Thanks again, Jo