Nu har även Wikipedia fått nys om saken...
The story tells of a green meadow located "this side of Heaven" (i.e., before one enters into it). Rainbow Bridge is both the name of the meadow and an adjoining bridge connecting it to Heaven.
According to the story, when a pet dies it goes to the meadow, having its body cured of any illnesses, frailties and/or injuries. The pet runs around and plays with other pets, missing only one thing – the love and companionship of its owner, who is still alive on Earth.
Upon the pet owner's death, on his/her journey toward Heaven s/he crosses the meadow. While doing so, the pet (along with any other pets the owner had while on Earth) spots its owner and runs to greet him/her. Reunited, the pet(s) and owner cross the Rainbow Bridge together into Heaven, never again to be parted.