Origins
To emerge is to break the surface. The word comes from Latin ēmergere — 'to rise out of' — composed of ē- (from ex-, 'out of') and mergere ('to dip, to plunge, to sink'). The image at the heart of emergence is aquatic: something submerged that rises into view.
Latin mergere produced a tight family of English words, all built on the idea of plunging. To merge is to plunge together — two rivers merging, two companies merging, both dissolving into each other. To submerge is to plunge under. To immerse is to plunge into. And to emerge is the reverse: to rise out of whatever you were plunged into.
Emergency preserves the original drama. An emergency is a situation that has suddenly emerged — surfaced without warning, demanding immediate attention. The Latin emergentia meant 'an arising, an unforeseen occurrence'. The urgency was built into the word from the start.
Old English Period
The word entered English surprisingly late — not until the 17th century — possibly because earlier English had adequate native alternatives like 'come forth' and 'arise'. But emerge offered something its Anglo-Saxon rivals lacked: the specific image of rising from concealment, of breaking through a surface.
In modern science, emergence describes the phenomenon where complex systems produce properties that none of their individual parts possess — consciousness emerging from neurons, weather emerging from molecules. The etymology fits: something rises that was not visible in the parts alone.