To go out. An exit is what makes a place a room rather than a trap.
Latin exitus, from exire (to go out), ex- plus ire (to go); the noun carried in Latin a further sense of an outcome or an ending, even a death, before English kept only the plainer sense of a way out.
In Latin the same word could also mean an outcome, even a death — the ending a path was always going to reach. An exit keeps only the gentler half of that old sense: not the end of everything, only the end of being inside.