If they are Jiffys, the manufacturer intended for them to be planted with the netting. The roots go through that.

I do, though it doesn't always come off easily. The roots dig into the pot so it takes a little work to loosen it up. If you find it a bit too challenging and don't care about ruining the pot, some will just replant with the old pot on. The roots can grow through.

You're using jiffys if I'm seeing your pictures right, right? If the roots have already grown into the fabric and you want to remove it, you could damage the roots. I would just plant the jiffys with the fabric into the new pot. The net will dissolve over time.