Burglar's wine is exactly what it says it is: poisoned/drugged wine to aid burglary. A servant accomplice, who had either been planted in the household or corrupted, would serve it to the master and mistress, and when they were unconscious would let the burglar in.

At least that's how it was explained in a Peter Lovesey novel that I read years before I first heard "Down in the Willow Garden", and whose title I have unfortunately since forgotten.