What is "rendered chicken fat"?
When heated above 200°F/95°C, the fat in meats will liquify and can be used for cooking.   This process is called rendering.  As rumtscho said, you're cooking the chicken in the fat released when it was heated the first time.

You can also use a long, low-temperature baking to render the fat out of duck carcasses and skin for use in duck confit.  In this case, the fat should be skimmed and filtered with a fine cheesecloth or fine chinois to remove bits of meat.

If you filter the fat from meat and cook off water, it will become very nonperishable, and can be used for cooking for a long time.  Bacon and duck fat are liquid gold for cooking, and can make even mundane dishes (roast potatoes) gourmet treats.