The phrase to let the dog see the rabbit means to allow someone to get on with the task that they are supposed to perform, without interference or restriction.

It refers to the fact that, when a dog sees a rabbit, it runs after it—as illustrated by the following cartoon published in the Yorkshire Evening Post (Leeds, Yorkshire) of Thursday 24^th July 1930:

Calamities—No. 12.

The dog sees a rabbit.

The earliest recorded instance of to let the dog see the rabbit in the Oxford English Dictionary (3^rd edition, 2010) dates from 1934, but I have found much earlier uses, which indicate that the phrase:
– originated in Wales
– with reference to fair-mindedness in sports.

The earliest occurrence of the phrase that I have found is from Football Notes, by “Old Stager”, published in the South Wales Echo (Glamorgan, Wales) of Monday 13^th March 1893:

Had the Seasiders’ played in their accustomed fine style, they would have made the game decidedly uninteresting, because all the scoring would have been on one side. As it was, they seemed to bear in mind the fact that they were opposing a team considerably their junior, and by checking somewhat their efforts—or rather, by refraining from any unusual earnestness—they let the dog see the rabbit and showed there was no ill-feeling in the matter.

The second-earliest instance that I have found explicitly indicates that the phrase originated in Wales with reference to equitableness in sports; it is from The Sporting Life (London) of Wednesday 17^th February 1904:

Everyone knows the keenness of the Welshman in the matter of sport, and few will have the temerity to say that they are not splendid judges of everything appertaining to it. The best is generally good enough for them, and if they get a run for their money they are generally satisfied. There is an old saying, and a trite one in the Welsh hills and mountains, “Let the dog see the rabbit.” This, literally translated, means that the Welshmen’s idea of fair play is that each side should have its 50 per cent. of the game, and as there is very little inclination to “gouge” among the sports of the hills and mountains, sterling good matches are ratified and brought off.

The earliest unambiguous use of the phrase in its current sense that I have found is from an article about the mock-parliamentary election at Loughborough College, published in the Nottingham Journal (Nottingham, Nottinghamshire) of Tuesday 21^st February 1922:

To-night mass meetings were held in the North and West Canteens.
The Labourites with a great display of red met in the former, with Mr. J. R. Williams in the chair. There was a large number of Tory and Liberal supporters and the early orators had great difficulty to make themselves heard.
“Give him a chance: let the dog see the rabbit,” was one of the interruptions.