The link was established in 1933, when researchers at London's National Institute for Medical Research infected ferrets with material from human influenza patients and reproduced the disease. One of the ferrets subsequently sneezed on a scientist, who developed influenza, closing the loop in the opposite direction.
The reason ferrets work is receptor distribution. The sialic acid linkages that human influenza viruses bind in the upper respiratory tract are distributed in ferrets much as they are in people, unlike in mice. Infected ferrets sneeze, run fevers and shed virus from the nose, and transmission studies place infected and naive animals in adjacent cages to test airborne spread.
This is why ferret transmissibility became the central question in the controversy over gain-of-function experiments on H5N1 avian influenza around 2011. Work showing that a modified strain could pass between ferrets by air was read as evidence of pandemic potential. The model has limits: ferret transmission is a strong indicator, not a guarantee of human-to-human spread.