c. 1300, "unfold, spread out, unfurl" (a banner, etc.), from Old French desploiir (Modern French déployer) "unfold, unfasten, spread out" (of knots, sealed letters, etc.), from Latin displicare "to scatter," in Medieval Latin "to unfold," from dis- "un-, apart" (see dis-) + plicare "to fold" (from PIE root *plek- "to plait").
Properly of sails or flags (and unconnected to play); meaning "reveal, exhibit, expose to view" is late 14c.; sense of "reveal unintentionally, allow to be seen" is from c. 1600. Related: Displayed; displaying.