Let us remember that the nursery that the children have devoted so much time to is telepathically tuned in with the thoughts of the people in it. Anything they imagine will be depicted on its walls. As we read the story and we see the hatred that the two children feel towards their parents because of the way in which they have become completely obsessed by the nursery. The rather disturbing scene of the veldt with its hungry lions pawing the ground and threatening to kill them in a rather life-like manner is one manifestation of their anger and hatred. Throughout the story, George and his wife hear screams that sound rather familiar coming from the nursery. However, it is only when they themselves are trapped in the nursery and scream that they realise why they are so familiar:
Mr. and Mrs. Hadley screamed.
And suddenly they realised why those other screams had sounded familiar.
The disturbing conclusion that these lines point us towards is that Peter and Wendy have spent hours fantasising over the deaths of their parents, achieving their revenge by imagining them being killed and eaten in this veldt by the lions. It is only at the end of this story when their desires become reality.
To answer your question, one needs to consider its context. The story imagines a futuristic world in which machines have been designed to meet humans' every whim and need. In this story, the family have bought a house which does exactly that.
George and Lydia Hadley had purchased a soundproofed Happylife Home and added a nursery which took the pride of place. There later appeared, however, to be problems with the nursery and they were concerned, so much so that they considered calling a psychologist to have a look at it. The Hadley's were the doting parents of two ten-year-old children, Wendy and Peter, and had had the nursery built for their benefit. The two children were spoilt rotten and resented it when their parents limited their privileges or freedom.
The nursery could respond to whatever thoughts or feelings were expressed by those who occupied it and would create three dimensional images of what was imagined. These scenes were very realistic since the room had also been designed to recreate every aspect of the environment which it depicted, including its smells, sounds and atmospheric conditions.
Lydia believed that there was something wrong with the room and when she and her husband visited it, they were greeted by scenes of an African veldt with lions in it. The lions seemed to have just been feeding and the couple was quite unnerved by their experience. They heard screams and George actually felt the heat of the African sun. It seemed as if the lions were going to attack them, so they ran out of the room.
George tried to change the scene in the nursery to another setting but it would not respond. When their children returned from a visit, they were told that the house would be shut off. They were extremely upset about this. Peter even threatened his father.
The psychologist arrived and advised the parents to turn off the nursery since it had become a medium through which the children were expressing their hatred for their parents. This deep resentment stemmed from the fact that the parents, on some occasions, had denied their children certain privileges. He also mentioned that the nursery had replaced the parents in their affections.
George also told him about the screams that they had heard emanating from the room, which the psychologist experienced this first hand during his visit. At the time, the children were asked to leave the room state and the nursery was then switched off. The children were devastated and begged their father to not be so cruel. Peter told his father that he hated him. Later, at his wife's urging, George switched it back on 'just for a minute.'
George and Lydia went to get dressed for the family would be leaving shortly. The children, who were in the nursery called to their parents to come quick. When the two got into the room, the children shut them in. They then witnessed the lions approaching them, ready to pounce and they screamed. 'And suddenly they realized why those other screams bad sounded familiar.'
The screams were their own. It seems as if the two children had succeeded in achieving the impossible. During their interaction with the room, they had trained the lions to attack their parents by imagining them and bringing them to life. This is evidenced by George, at one point, finding his damaged and scratched wallet and Lydia's bloody scarf in the nursery. Added to this were the repeated screams. Peter and Wendy had succeeded in transcending the illusion and turned it into a reality. The lions and everything else was not just realistic, but real. When George and Lydia were trapped in the room, they were actually attacked and devoured by the lions.
When Mr Mclean, the psychologist, arrived later, the parents were gone and he could see, in the nursery, that the lions had just stopped feeding and were going to a waterhole. It is obvious that he would not be able to explain the parents' disappearance because, ironically, when George raised the question of the room actually coming to life earlier, he pertinently dismissed his suggestion.
The two malicious children had found a perfect way to get rid of their parents.