Haran suggests the bars were adopted from Rome. The second bar was added to avoid having to unwind and rewind the scroll at every use, and was used in Rome by at least the first century CE.
Based on literary sources, Haran argues that a single bars was used for all scrolls until near the end of the Talmudic era, when double bars became standard. There is no recorded reason for the change and it is likely that it was done for pragmatic reasons.
See "Torah and Bible Scrolls in the First Centuries of the Christian Era", Shnaton 10:100-101. (In Hebrew.)
