The “fell” here means that the person fell. They fell “over the bucket” which means the bucket was in the way and it was the reason they fell. It’s a lot more common to hear “tripped over” something rather than “fell over” in this situation. If it was referring to the bucket, they would use “knocked over” or “tipped over” or something like that.

"I tripped and fell" means that my foot collided with some object and I lost my balance and ended up on the ground. "Over the bucket" indicates that it was bucket that got in the way of my foot, and I went over the bucket as I headed towards the ground. It normal to expect that the bucket fell over and spilled its contents when it was hit by my foot.