Mr. BrownThumb explains it all

clipped from mrbrownthumb.blogspot.com

If you’ve grown a daylily chances are that you’ve noticed a new plant forming along the scape (flower stem) of one of your plants. The other day I came across one growing on a browning stem of one of my Daylilies. Hemerocallis growers call these plantlets that sprout from the stems “proliferations.” Sometimes a proliferation can grow enough during a growing season to actually flower, most of the time they’ll grow just enough for them to be harvested and planted. Proliferations are exact clones of the plant they are growing from and they’re a great source of free plants.