Hello, and thank you for a using Ask an Expert to learn about propagating your daylily.
What you have is called a proliferation, “prolif” for short. A prolif is a new plant growing on the flower stem and is an exact copy of the original dayily. If you look closely you may see new roots already beginning to develop. I usually wait until I see roots about a quarter inch long, then cut the flower stem about a half inch or so below them. (That bit of extra stem helps stabilize the “cutting” in the pot. You can trim off all the flower stem above the prolif.) I suggest you root the prolif in a small pot filled with a moist growing medium, but not soil.
You’ll find detailed instructions how to proceed in “Take advantage of multiplying daylilies’ http://msucares.com/news/print/sgnews/sg01/sg010827.html You’ll need to protect the prolif through the winter as it develops roots then, after the soil warms, plant it into the garden.