튀어나오다 is something shooting out or sticking out — eyes bulging, a bone that has protruded since birth. Of speech it is words coming out before you can stop them, and 나도 모르게 튀어나오다, blurted out without meaning to, is what two captions here are labelling. The -게 slot gives the manner: 자연스럽게 (naturally), 두드러지게 (prominently).