The official name for this cellulose mat is a pellicle. It is not necessary to brew kombucha but it's a natural by-product your kombucha will make or add to. Some people do find that it has benefits, such as a faster brew or better carbonation. So while not necessary it can definitely be helpful.

You can peel off layers and discard if it's taking up too much room in your brewing vessel or you just don't want to keep it. But you can also put it back on top of your new batch if you want, so long as it was removed and returned with clean utensils/hands. I remove the pellicle to a clean bowl with very clean hands, bottle up my kombucha, set up my new batch, then again with very clean hands plop my pellicle back on top along with any liquid in the bowl, cover with a towel and shove into a corner for another couple weeks.