Leaving in the middle of the night with all of your stuff without paying your landlord or creditors, in turn of the century jargon. Flitting day was the traditional day of the year when everyone's lease was up and therefore everyone who needed to would move on that day. Moonlight-flitting would obviously take place before this date (which was usually around May 25th). This was perhaps made significantly easier with the advent of things like the fire escape.