The expression refers to using your senses to make determinations rather than depending on mechanical aides. You might hear someone say something to the effect, “I’m not certain what I’m doing, but it feels right.”

 According to Tony Kleu, pilots in early airplanes had no instruments to help them turn the plane and stay in control. They relied on their sense of balance and feelings transferred to their bodies by contact with the seat to determine the aggressiveness of the turn. A hard turn would cause them to slip in the seat. Thus “flying by the seat of your pants” is literal.