Take your cue

And from the same book: "He sounded emotionless and professional. Jackpot too his cue and shut his mouth."

'Took his cue' obviously means took the hint, got the message, etc. But why 'his cue'? Is this from the theatrical term? And if so, why a cue anyway?

Yes, the theatrical term. American Heritage Dictionary: "Cue . . . A word or bit of stage business signaling the beginning of another action or speech . . . [Origin obscure.]"