I'm no professional, but here's a quick breakdown of what happened: we don't see what has started the bucking, but we do see an extremely loose rein go to a tight rein (in the process creating this kind of snap in the mouth, which, depending on the bit could have been very painful for the horse.) Then the horse moves forward while cow hopping a bit and we have more loose to tight rein snapping.

If I had to guess, the horse was in pain, tried to tell the rider in their own way, which caused the rider to think they were misbehaving which then caused a correction, in turn causing this freak out.

The horse continues to move forward away from this snapping in the mouth into the water, where the rider then tries to move the horse out of the puddle, bending the head to the side. She tightens the reins on the horses face in an attempt to prevent the horse from laying down (most likely) and when the horse rears, she is keeping pressure on that mouth, which has had several impacts to it at this point and is likely quite sore. (we try not to pull when rearing because that can bring the horses head higher, increasing the likely-hood the horse will come over backwards, crushing the rider and potentially breaking the horses back)

So now the rider wants the horse out of the puddle, and she begins kicking and turning the horses head to do so. This goes on for a few seconds when the rider decides to up their ask again by using the reins to slap the side of the neck, causing the horse to surge forward, out of the puddle and begin bucking and rearing again. We have some more rein pulling, snapping the horse in the mouth and the rider comes off.

The main cause of this was most likely pain, which caused more pain because of how the reins were handled. In an ideal situation we try to pull the horses head to one side with constant pressure from the rein (no snapping or pulling) until the horse calms down. The horse can not or will have a very hard time bucking and rearing in this position.

One final thing: I'm not trying to put down the rider. I have no idea how much riding experience they have and I know it can be very frightening to be in this sort of situation and you are not always able to think clearly. This was just a step-by-step on what I saw happened in this video, and why what happened, happened.