Something "essential" is something pertaining to the essence of a thing, and so what Simon imagines is an illness (not necessarily a sickness, the state of being ill, but perhaps a badness, an evil, a nastiness) which is at the very heart of all humans. What Golding's narrator describes Ralph, in the final page of the book, weeping for is "the darkness of man's heart".
The natural evil of humans, then, is something inexpressible: and indeed, the quote above comes from the sentence "Simon became inarticulate in his efforts to express mankind's essential illness". This darkness, this evil, is something difficult to pin down in words: it is the invisible moral weakness in humans that allow the events of the novel, and its descent into savagery, to occur.
So "mankind's essential illness" is not actually Simon's phrase, but Golding's. What Simon is trying to express is that there is a beast (or something, if you like, beastly, on the island) and that, as he says, "maybe it's only us". Simon's meaning is that the beast - the destructive force which scares the boys - is perhaps not an external force, but something dark within the boys themselves: not in a supernatural way, but something fundamentally bad present in human beings.