To give a little background to what others have said: This is based on a sports metaphor.
To field a ball in a sport such as baseball (which is played on a baseball field) is to catch it or chase after and pick it up -- IOW to get hold of it. Fielding balls is a way of practicing this: balls are repeatedly thrown or batted, and you field (chase after) them as quickly and accurately as you can.
From that literal fielding came the idea of catching and handling any number of things that are "thrown at you" -- in particular questions. Someone fires questions at you quickly and you answer them quickly and spontaneously. It's like fielding balls.