I would recommend reading these documents provided by Peter Langman.[You must be registered and logged in to see this link.][You must be registered and logged in to see this link.]
Officially the motive remains 'inconclusive' however here is some (a lot) information if you're wondering:
One of the most important things to look at when attempting to comprehend Lanza's motivations are his views on civilisation and humanity. Many consider Lanza to be an anarcho-primitivist who vehemently opposed culture, civilisation and in the end, humanity. Lanza called into anarcho-primitivist John Zerzan's radio show 'anarchy radio' a year prior to the shootings, where he expressed his thoughts on a domesticated chimpanzee named Travis who ran amok following a life in which he spent his time "enjoying" the products of civilisation. Lanza expressed sympathy towards Travis and if you pay close attention to Travis’s story you can tell that much of what happened in his life was similar to what Lanza experienced.[You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.]
With this information gathered it can be safe to assume that some of Lanza's possible motivations could be rooted in a disdain towards how children are 'indoctrinated' into civilisation, and him killing them would hypothetically be saving them from the horrors of society because of how it impacts those who live in it.
Lanza had also expressed sympathy towards pedophiles who he saw as unjustifiably ostracized in society, he even wrote a 30 page essay about it and his opinions regularly slipped into his online writings. Lanza destroyed his hard drive before the massacre, however authorities were able to retrieve writings he had made on a seperate hard drive where he wrote screenplays and essays about pedophilia. One of Lanza's former psychiatrists plead guilty to sexual assault a few years back and this is notable because in his online writings he refers to therapists as "The Rapists". In one of the documents found on his hard drive he also claims that he was molested by medical professionals when he was a boy. It's possible that if true, these could have contributed to his obsession with pedophilia because of his exposure to it as a child.[You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.]
If Lanza was a pedophile it's possible that his motive behind the shooting could've been rooted in a sadistic sexual fantasy, this is supported by the eyewitness accounts which described him screaming at the children before killing them. Despite the evidence which paints Lanza as a pedophile, he repeatedly denied this, however this could be a lie in order to not face the discriminatiuon which pedophiles face (which he was well aware of).
As to why he specifically targeted children, my guess would be because of Lanza's emasculated and frail state at the time of the shooting. You have to consider the fact that he was 114 pounds and 6 feet tall, at some points he reached a dangerously low BMI of 14. If Lanza were to attack a place where more adults were present he would've been easily tackled and his master plan would be foiled. His obsession with death as well as plans to commit suicide would lead to him being anxious if even the slight possibility of failure was present, so he picked the absolute weakest target in order to not be thwarted.
It was concluded that his anorexia had gotten to a point so severe that he suffered brain damage from malnutrition. This coinciding with his already present mental illness could have fuelled his drive to kill, the more days that passed the worse his mental state became. This finally culminated into him being on the verge of death from his illness, and he saw no other option than to end his life, but not before taking others with him.
Of course this is only speculation but I thought I would give my two cents on the matter. I may have not explained all possible motives but I don't feel like writing a whole thesis on the matter right now. I apologise for any spelling errors as I am writing this before I go to sleep.[You must be registered and logged in to see this link.]