It may be surprising, but the material implication does not come from truth tables, the truth table definition is a late development. Neither de Morgan, nor Peirce, nor Frege, nor even Russell came up with it or justified it by matching Boolean operations to something in Plato and Aristotle. A detailed story can be found in Cajori's History of mathematical Notations, vol. II.
It came from a very common idea of classical logicians of identifying propositions with classes (intensions), and classes with sets (extensions). Accordingly, the early definitions of implication interpret "X implies Y" as "X is contained in Y". Originally, it was only applied to syllogisms, where it matches the intuition.
In the 19th century, its scope was expanded with the algebraization of logic by Boole and de Morgan. The chain of transmission went from de Morgan's
) (1847, same year as Boole's first treatise), to Peirce's claw ―< (1867), to Schröder's $\supset$ (1890), and Peano's Ↄ. Later Peano, and after him Russell, adopted Schröder's $\supset$ (note that the meaning is reversed compared to the modern set inclusion). Peirce called the material implication "the copula of inclusion" (also "illation"), and Frege (whose notation for it was clumsy and never reproduced later) even criticized Schröder for "confusing" it with class inclusion.
The identification itself predates Boole and even Leibniz, it can be traced back to Aristotle, and was implicit in scholastic logic (for syllogisms). Russell's Principia still has a trace of it, the class/set identification only goes away after Hausdorff's Grundzuge der Mengenlehre (1914), see Kanamori's The empty set, the singleton, and the ordered pair.
Of course, X is not contained in Y if and only if there is something in X which is not in Y. Frege and Peirce understood this truth functional consequence of the proposition/class identification, and made it definitional when transitioning to a logic with quantifiers. For example, Peirce wrote in 1883 (quoted from Dipert, Peirce's Propositional Logic):
"To say that an inference is correct is to
say that if the premisses are true the conclusion is also true; or that every possible state of things would be included among the possible state of things in which the conclusion would be true. We are thus led
to the copula of inclusion".
Frege's work remained buried until Russell brought it back from obscurity in Principia. The rest (including Peano and, through him, Russell) adopted notation and conventions of Schröder's Algebra of Logic, which followed Peirce, where an equivalent of $A\supset B=\lnot A \lor B$ already appears, see Dipert Peirce, Frege, the logic of relations, and Church's theorem. But Peirce used truth tables only sporadically and in unpublished manuscripts (1893 and 1902), so they did not become common until Russell and Wittgenstein reinvented them in 1912.
So the material conditional gradually emerged from a cluster of intuitions about propositions, classes and sets. But there are only two cases where it fully applies in its modern form:
- Conceptual containment in syllogism (a la Aristotle and Kant). This form is too narrow to cover our intuitive notion of inference.
- The model-theoretic definition of extensional entailment in modern mathematics, a.k.a. semantic consequence, a la Tarski. This model does not entirely match the intuitive indicative conditional. Hence the cognitive traps:
"The material conditional allows implications to be true even when the antecedent is irrelevant to the consequent. For example, it's commonly accepted that the sun is made of plasma, on one hand, and that 3 is a prime number, on the other. The standard definition of implication allows us to conclude that, if the sun is made of plasma, then 3 is a prime number. This is arguably synonymous to the following: the sun's being made of plasma renders 3 a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another...
...Another issue is that the material conditional is not designed to deal with counterfactuals and other cases that people often find in if-then reasoning... A further problem is that the material conditional is such that (P ∧ ¬P) → Q, regardless of what Q is taken to mean. That is, a contradiction implies that absolutely everything is true."
An interesting reconstruction of how truth functional connectives became implicit in the vernacular of mathematical proofs is in Azzouni's paper, pp. 37-38.