There is a common phrase: "look what the cat dragged in!". It is a jocularly dismissive comment about e.g. someone who has just entered the room.
Cats tend to "drag in" birds and small rodents. ("Drag in" means "bring in", with an implication that the thing brought is not entirely willing to be brought.)
In a restaurant where cats wait on cats, and serve food presumably cooked by cats, the "chef's special (dish)" might literally be "something the cat dragged in" (i.e. a bird or small mouse).
Thus the humour lies in taking a metaphorical phrase and presenting it as literal.
(I feel my explanation may well have killed the very small residue of humour the cartoon may have had.)