brief (n.) from L. breve (gen. brevis), noun derivative of L. adj. brevis (see brief (adj.)) which came to mean "letter, summary" (specifically a letter of the pope, less ample and solemn than a bull), and came to mean "letter of authority," which yielded the modern, legal sense of "summary of the facts of a case" (1630s). The verb meaning "to give instructions or information to" (1866) was originally "to instruct by a brief" (1862); hence briefing, first attested 1910 but popularized by WWII pre-flight conferences.

[Dictionary.com]

As a member of the legal profession, all I have to add is that "summary of the facts of the case" isn't quite right; better would be "summary of the law that applies to the facts of the case." The term passed from ecclesiastical law to civil law because the applicable civil law is an authority on what a judge should do in much the same way that a papal proclamation was thought of as an authority on what Catholics should do.