That looks like a spidermite web. They're very harmful to most plants they invest causing the leaves to wilt and fall off as described.

Thank you. I will now try to find a remedy.

Typically, lack of humidity is the culprit. Spider mites like it warm and dry......the usual conditions indoors. I have no idea how big your fig is but hosing or spraying it down with water periodically (like once a week) will help. So will spraying with a horticultural oil like neem oil. Neem oil is a particularly effective miticide.

Take it outdoors. Hose it off, especially the undersides of the leaves. Repeat every several days.