The way to apply the quotation from Mark Twain to Golding's novel is to think about how it expresses an underlying theory of human nature. Among the two competing philosophical theories of human nature most relevant to this issue are those of Thomas Hobbes and Jean-Jacques Rousseau.
According to Hobbes, especially in Leviathan, human nature is inherently selfish and chaotic, and only a strong government can keep in check the natural human propensity to self-centered behavior that harms others such as robbery and murder. Hobbes sees children as savage beasts in need of civilized restraint by the authority of parents and government; essentially his account is almost a Calvinist one, without the redeeming influence of God. Rousseau, on the other hand, sees children as inherently innocent and virtuous, and contact with adult civilization as corrupting them.
Lord of the Flies is almost a fictional proof of Hobbes' account, arguing that when freed of civilized restraints and of the strong hand of authority, children reveal their dark inner natures, including irrational fears and hatreds, bullying, rage, tribalism, and other primal desires, and act in an unspeakably evil fashion. The dislocation of the children from ordinary British life and absence of adult authority allow them to reveal this dark side.
This quote can be related to Golding's Lord of the Flies because all the children there show this "darkness" within them. It starts with their arrival to the island and them immediately taking all their clothes off. They are slowly digressing to show their true colors. As the story continues on you start to observe people slowly cracking and eventually releasing their darkness. This is most apparent during the death of piggy.