I'll have a bash, but there'll be others better at describing this for you. You want to try to meet an obstacle - whether you're just putzing around at home, in a showjumping arena, or hurtling towards a downhill hurdle on board a wayward novice - in a manner to produce the best possible jump for what you're doing. That sounds daft, but if you can't see the stride coming up for yourself, it's best to let the horse fiddle itself over as best it can. When you're pretty sure you've got that magic last three strides into the fence right, you can push on, since the horse only wants your support. When you think 'oh, feck, I'm not sure whether to ask for a long one or shorten him up', it's best to leave well alone. A professional horse will sort it out for you, 9 times out of 10, and save your blushes. Niggling and faffing away will simply, literally, put him off his stride, so sit still, and wait. That often means a monster leap like that by the wonderful, but thoroughly experienced, MISTER MCGOLDRICK, soaring over from a clear stride away from a normal take-off point, or else getting in a bit close and coming up with a diddly little one past the point of optimum take-off. That tends to lose a bit of momentum as the horse has to rise more steeply to take the jump, and, in an inexperienced or harrassed horse, can see them pitch on landing.
The take-off point for a horse is where it can make the best possible shape over the jump, clear it without mishap, and land in a position to immediately take up running or getting ready for the next one (in the case of showjumping). The optimum shape over a chase jump or show jump is a parabolic curve (or just plain parabola), from take-off to landing. If your horse has short, choppy strides, he's likely to make a much less graceful parabola than a long-striding one, where it can be quite long and low. The importance of 'shape' is that it determines how fluently the horse incorporates the obstacles into his galloping - if he consistently won't obey instructions to attack his fences, such as always shortening up before take-off, we all know the result - there are breaks in the momentum and he can easily be outjumped in the air by a bolder horse. The better the fluidity of the arc over the jump, the better the ability to keep going on t'other side.
How lovely this sight is, in chasing, depends often on the going: we've been seeing a lot of rather triangular shapes of late, where desperately-tired animals have been lifting themselves almost vertically, because they're too tired to make a bold leap over the jumps, and coming down sharply, just about able to canter on. When they're so tired, some can't get a front lead leg out fast enough, resulting in a tired, pitching fall.
Conversely, hurdling over Good ground often sees a flying take-off, a flat shape through the air, and a very low landing point, yards from the obstacle. Most hurdlers don't need to make a 'good' shape since they're not required to set themselves up for hurdles in the way that chasers need to set themselves to clear their much heftier and far more solid jumps. The booting into hurdles usually comes at the end of the race, encouraging the horse on, with little real attempt to 'present' to the obstacle.
I would say that when you say you see a horse looking likely to fall, be aware that the jockey is probably even more painfully aware! You do get a sense of the rhythm going wrong, or a headstrong/naughty/nutty horse just not being respectful, and that is why the jock will 'just sit there' - there's no point encouraging an even faster cartwheel!
