The role of the epiglottis in the respiratory system is to act as a protective flap that covers the trachea during swallowing. This ensures that food and liquids do not enter the airway, directing them to the esophagus instead. When breathing, the epiglottis remains open, allowing air to flow into the trachea and subsequently to the lungs. This function is vital for preventing choking and ensuring efficient gas exchange during respiration, maintaining the overall health of the respiratory system.