To expand on Maxey's excellent comments, "off with you" is an archaic, old way to say "scat", an imperative command to "go on!" or "get out of here!" So the speaker is saying "get out of here and go to bed!"

Short for "it's off to bed with you." It's just one of those phrases that got into the language for which there is no logical reason. It makes sense if you phrase it as: "The situation WITH YOU is now OFF TO BED" = What is going to happen now is that you are going to bed. It's used by parents for emphasis when they want to tell a child to go to bed, but don't want to make it a harsh command.