Please note that “Sophistry” is a modern derogatory term for Rhetoric applied in a particular manner, and is not necessarily representative of the beliefs of all Sophists, except that they generally taught Rhetoric.
In traditional argument a set of premises are connected together according to the rules of logic and lead therefore to some conclusion. When someone criticizes the argument, they do so by pointing out falsehoods among the premises or flaws in the logical scaffolding. Generally some judge or audience concurs with or rejects the criticisms, and thus the truth is eventually arrived at.
The essential insight of Sophistry is that it is not the solidity of the argument but the rulings of the judges which ultimately determine whether a conclusion is considered true or not. By appealing to the prejudices of the judges, one can garner favorable treatment for one's side of the argument and cause a factually false position to be ruled true.
The philosophical Sophist goes one step beyond that and points out that since it was traditionally accepted that the position ruled valid by the judges was literally true, any position ruled true by the judges must be considered literally true, even if it was arrived at by naked pandering to the judges' prejudices — or even by bribery.
The modern peer-reviewed Journal is exactly reflective of the traditional model of argumentation. Alan Sokal demonstrated both with the Sokal Affair and at length in his book Fashionable Nonsense that peer-review is not necessarily protection against falsehood entering the canon of accepted truths.
Sophistry (Rhetoric) is a perfect example of a disease meme. It is successful because its core insight is literally true: if truth is determined by the rulings of corruptible humans, then truth can be changed by corrupting them. Any society or individual that searches for truth must be on guard against the influence of flattery, comfortable assumptions, and pretty phrases. It is a disease meme because once it has achieved dominance in an organization the organization can no longer function effectively, and fails.
There is no sense in which phrasing an argument well makes it more true; therefore there is no sense in which Rhetoric brings us closer to truth. Instead the purpose of Rhetoric is to make things appear to be more true (acceptable) than they actually are: and so it leads inexorably to the acceptance of falsehood.