The dalmatic was ordinary clothing before it was anything ecclesiastical. It takes its name from Dalmatia, arrived at Rome as a fashion in the imperial period, and was a wide-sleeved tunic of the sort a well-dressed Roman might wear, decorated with two vertical stripes called clavi running from shoulder to hem. Roman moralists disapproved of it as loose and foreign, which is the usual fate of an imported garment.
It became a vestment by being a privilege rather than by being prescribed. At Rome it was worn by the pope and by the deacons of the city as a mark of honour, and popes granted the right to wear it to particular churches and particular deacons elsewhere as a favour, so its spread across the west can be tracked through the grants. Only when the grants had accumulated far enough did it become simply what a deacon wears.
The stripes survived the transition, sometimes as woven orphreys and sometimes as applied bands, and the shape settled into a knee-length tunic with wide open sleeves, matching the colour of the season and normally matching the celebrant's chasuble as part of a set. Its Byzantine counterpart is the sticharion worn with the orarion, which is a different shape doing the same job.