A Freudian analysis of Lord of the Flies suggests that "man's essential illness" is caused by an unbalance of the psyche, more specifically that the id has become too dominant. Freud's model divided the human psyche into three distinct sections: ego, superego, and id.
The id represents man's more base instincts, like libido. In Lord of the Flies, Jack represents the id. His obsession with the hunt and the raw, natural power it provides quickly devolves him from civilized choir boy to animalistic savage. As Jack gains power on the island through the approval and support of the other boys; civilization declines.
On the other hand, Piggy represents morality, perfection, and the conscience; he becomes the embodiment of the super-ego on the island. Piggy speaks consistently to doing the right thing; many times he provides the voice of reason to distill the boys' fears about the beast or to prompt the other boys to focus on rescue and returning to civilization. Piggy, however, is extremely weak; he cannot stand up to Jack's savagery or bullying ways. Like in the Freudian model, Piggy also needs a mediator; this is where the ego intervenes to create balance between the self-serving needs of the id and the altruistic wishes of the super-ego.
In Lord of the Flies, Ralph acts as the ego. He mediates and attempts to resolve differences of opinion between Jack and Piggy, but Piggy is so weak, most of the time Ralph must intervene on Piggy's behalf.
Golding's novel Lord of the Flies ultimately suggests that unchecked id leads to destruction and savagery. Man's essential illness is his struggle to contain or muffle the id, in favor of the more noble super-ego. The boys in Lord of the Flies are unable and ill-equipped to maintain their ties to civilization; the resulting mayhem and deaths are a direct reflection of the harmful effects of man giving into his baser instincts.