Some people are unclear on the reasons for fining a wine. They get it confused with filtering.
As discussed in an earlier post, filtering helps get the bugs out and improves the clarity. It may even alter the texture of the wine a little.
Fining works in conjunction with filtering. Fining involves mixing or blending in a substance that is intended to act on an aspect of the wine and change it's overall sensory perception.
Fining agents don't generally remain in the wine. They fall to the bottom of the tank and the clear wine is 'racked' off or they are left behind after the next filtering session.
The practice of fining is very old. Early on it was discovered that things like blood and egg helped smooth out a wine and made it easier to drink at an earlier age. These days there are a myriad of sanitary agents (some still based on those old originals) that a wine maker can trial to achieve just the right effect.