The suggestion surely is that the animal has been thrown down - as a result usually of the swaying motion of the horse-box it has been travelling in - rather than that it is just lying down.
The OED provides this : Cast:(of horses ) "broken down, rejected as unfit for service" as well as the usual meanings.Possibly we used to think of a horse which is down and can't get up as having cast itself, landed,. placed, dropped, itself, with some force.
Well, the verb cast - and hence its past participle/adjective form - means throw(n) rather than just dropped. I'd say that's the meaning, whether it threw itself down or was thrown down by some external force, and not that it is just lying down.
I'd also add that I don't believe "won't get up" really applies. Surely that suggests that it could if it wanted to. Even the Horse and Hound link specifically says "unless it is cast".
It means that the horse has laid down in the box and is unable to get up, usually because it has laid down too near a wall and is either stuck against the wall or has injured itself trying to get up. In any event it is therefore unfit to race either through injury or through exhausting itself trying to get up.