Why is there such a big focus on children in Ender's game and why are only children trained in the battle academy? Why can't teenagers or adults start training also if they have whatever skills the academy is looking for.
While the accepted answer was somewhat correct (one of the 2 reasons was indeed the fact that you could "fool" a child into thinking they had been playing a game, so that they would be able to make the necessary tactical sacrifices without worrying about losing men they command; and not carry around the psychological burden of "what if we lose"), that was not the only reason, and possibly not even the main one.
The reason was articulated by Graff even before Ender was born, in First Meetings collection, in The Polish Boy, when he explained why he did not care that he wasn't getting John Paul Wieczorek (who we later would know as John Wiggin, Ender's father) into Battle School: it was because their research suggested that pre-adult kids were better commanders than grown ups.
"There is a definite fall-off in outcomes after the trainees reach adulthood," said Graff. "That's a fact, however much we may not like the implications."
"They know more, but do worse?" said Chamrajnagar. "It sounds wrong. It is hard to believe, and even if we believe it, it is hard to interpret."
... and further on, Graff expands on it:
"You're forgetting the research we've been conducting. It may not be final in some technical scientific sense, but it's already conclusive. People reach their peak ability as military commanders much earlier than we thought. Most of them in their late teens. The same age when poets do their most passionate and revolutionary work. And mathematicians. They peak, and then it falls off. They coast on what they learned back when they were still young enough to learn. We know within a window of about five years when we have to have our commander. John Paul Wieczorek will already be too old when that window opens. Past his peak."
Spoilers for Ender's Game below.
Because they probably can't fool teenagers or adults for long enough for them to destroy the buggers while believing that it's all still a simulated war game. And if they know it's real, then they will break under the pressure, and possibly also start to worry about committing genocide.
Since I'm not familiar with any official answer to this, I'll give another hypothesis.
The initial ships were launched years before the start of the book, around the time of the previous war ending. They were likely populated with adults who fought in that war. Thanks to time dilation, they do not age significantly, and are still capable of fighting upon reaching the buggers' world, though it's not clear if they can come back.
For those left on earth, there would be quite a wait, so it made sense to start training children in the hope that the right commander would come along. If that was to happen, they could then "preserve" the child(ren) using relativistic trips.
My impression reading the book was that the battle school system had been going on for a long time, but that the right candidate has not been found / engineered to that point, making Ender and his friends "the last hope" as the fleet was getting close to the bugger homeworld, thus the rush to advance the training.
Separately from that, there seems to be a common theme in many fantasy/sci-fi books of giving children out early to the organizations that foster them to a particular destiny. Makes them much easier to indocrinate as well. For example, the Dragon Age IP (while a game) has that notion with the templars and the mages.
I'll add to the "accepted" answer. The need for child training was two fold:
"Deception" was needed to cater to the "empathetic genius". If you love the enemy so much that you can understand and know their moves before THEY do - kinda hard to kill them all dead without hesitation. THAT, IMHO, was the deciding factor on using genius children. Adults who know what they are doing will make different decisions when they have to look in mirror and know they just sent someone... someone's father, brother, mother, etc... to their death. They wouldn't be able to make the decisions that needed to be made - out of love for their brothers AND their enemies.
Also... add to the fact that training leads to graduation... graduation leads to captains and generals of the fleet. A glorified "West Point" academy that starts training at a younger age to further instill the needed qualities in the warriors that make the war machine.
If I recall correctly, at some point it's heavily implied in the book that children are still young enough that they're willing to reach new conclusions and change their thinking if necessary, where adults are more set in their ways.
The big advantage with that is that if you start young, and start in zero-g, instead of on Earth, they'll be able to form new strategies that someone who's formative strategies required gravity, and were fairly close to two dimensional (see also Star Trek: the Wrath of Khan, where Kirk bests Khan because Khan couldn't think in three dimensions).