I have a fig tree indoors for the winter and I noticed on most of the leaves, there appeared this really fine web like structure all around the edges of the leaves. It doesn't look like a typical spider web because it is so fine and you can barely see the individual strands. Does anyone know what causes this? Is it some insect or something the plant produces itself?
There is lots written on treating spider mites. Whatever you decide to do, you need to repeat the treatment, I think every few days, until there are no new mites creating new webs. But if you you have other indoor plants, you also need to keep looking them over carefully, as very often the mites will infect more than one plant. If you're hoping to have edible figs, you could consider constantly going at the webs with a cloth and water (after a shower in the tub) rather than treating them with a poison.
Interesting, thanks! Are they harmful at all to the plant? Do they stay in the pots, or do they roam around the house?
Yes, they are harmful. There is a lot of information about spider mites and how to fight with them in the Internet. For instance: How to Get Rid of Spider Mites Effectively (Complete Guide)
I'm not sure if the damage would extend to killing the plant, but they do damage the leaves, which would affect the nutrition the tree gets and affected plants get to look really unattractive. The mites are very tiny, too small to see if they're roaming around the house, but they do get themselves from one plant to another. Here what seems like a good article: How to recognize and manage spider mites in the home garden | OSU Extension Service Edited - Sulev's posting got in while I was typing mine - I did not see it before I posted my link. I am not suggesting my link is better than his.