Your original Asiatic lily produced new offset bulbs - each of which has produced a stem and growth tips. Simply, now you have an Asiatic lily "clump".

After blooming and all the stems and leaves have browned out naturally, you might want to carefully lift the clump and separate the bulbs and replant where wanted. Early fall is a good time to do this.

Wow! Your bulb was very happy, to have reproduced like that! When you dig them up, I'd suggest a pitchfork rather than a shovel. And put it in a good distance from the clump, as some of the stems might not be coming straight up from the bulbs. Lift the entire ball 'o bulbs, then rinse off the dirt so you can see what you are doing. Then pry off the bulbs using your hands. REplant and /or share the wealth.