All my answers to this are flawed. My best is: It's like Euclidean geometry: humans (and other species) are constructed in a way that Euclidean geometry fits fairly well. The formalized rules of Euclidean geometry match spacial reality even better than what we've evolved, so we prefer them... but they're similar enough to what we've evolved that we accept them rather than alternate geometries. Euclidean geometry isn't right - reality is more complex than any system of geometry - but the combination of "works well enough", "improves on our evolved heuristic", and "matches our evolved heuristic well enough" combine to give it a privileged place. Just so, that system of formal logic works well enough, improves on our evolved reasoning heuristics, and yet matches those heuristics well enough... so we give formal logic a privileged place. The privilege is sufficient that many believe logic is the basis of Truth, that many theists believe that even angels or deities cannot be both A and not-A, and that people who use fallacies to convince others of truths are frequently considered to be liars. This does not sufficiently satisfy me.

An alternate answer, that a believer in absolute morality or logic might like, is that logic actually deserves a higher place than Euclidean geometry. Where geometry can be tested and modified wherever the data support a modification, logic can't. No matter how many times our modus ponens does worse than an Appeal to Tradition or Ad Populum in some area of inquiry, we still don't say "ok, alter the rules of logic for this area of inquiry to make Ad Populum the correct method there and Modus Ponens the fallacious method there", we just question our premises, our methods of detection of answers, etc. So logic is special and is above the empirical method. I am unsatisfied by the above paragraph as well.

A third possibility is that it's not - it's just a code of conduct/signalling. We agree to only use logic to convince one another because it works well, because the use of other methods of persuasion can often be detected and punished, and because the people who can rely on logic rather than on other methods of persuasion are smarter and more trustworthy. In specific instances, logic might not be the best way to learn something or to convince others, but getting caught supporting or using contraband methods will be punished so we all use/support logic unless we're sure we can get away with the contraband. This is an unsatisfying explanation to me as well.