Your plant is an Echeveria of some kind. Sorry, but I'm terrible at IDing the specific name. Those "offshoots" are inflorescence, or flower stalks. Quite normal.

I see some leaves lying around in the picture. Were you going to try to propagate those? Should work nicely. But the nice surprise is the leaves on the inflorescence. Those propagate quite well and more quickly than mature leaves! Be sure to grab those, even the smaller ones.

I'm sure more advice is headed your way.

Thanks for your reply! Should I cut down the flower stalks to a size that the plant can support? The weight of the stalks was uprooting the plant, wasn't sure whether to prop them up or cut them off.

If you want to cut flower stalks, cut them all off. Why just cut them shorter? - they won't bloom that way anyway...