In medieval logic, a disinction was made between material and formal consequence.
" John is a pianist, therefore John is a musician" : a material consequence, for the validity of the consequence depends on the semantic content ( matter) of the terms involved; it is not the case that, formally, " J is a P" implies " J is an M".
" All pianists are musicians, therefore, those who are not musicians are not pianists" : a formal consequence; the reasoning still holds when you replace the particular terms " pianist"/ " " musician" by variables.
A " consequence " ( in the medieval sense) is a reasoning ( with a " therefore" in it). This is not the same thing as an implication , whch only has an " if ... then " in it ( psychologically, an implication is a judgment, an assertion, not an inderence).
However there is an analogy between " material consequence" and " material implication".
A material implication only holds ( when it holds ) in virtue of the factual truth values of the propositions involved in it ( in the same way a material consequence holds in virtue of the semantic value of its terms).
Note : he truth value of a proposition is its semantic value understood as its denotation.
" If Biden is President of the USA then Harris is Vice-President".
This is a true material implication simply because the antecedent is true and the consequent is also true.
Saying that ths implicaion is true does not mean that Harris has to be Vice-President in case Biden is President. It simply means that, as a matter of fact, it is the case that both sentences are true, meaning that it is not the case that the first is true while the second is false.
Logical implication is different from material implication. The proposition " if P the Q " logically implies " if not-P, then not-Q", whatever the truth value of P and Q may be.
As to the question " why is P--> Q equivalent to ~P v Q?" . One could simply say that mplication is a derived connective, a connective that is defined in this way using primary connectives , namely negation and disjunction. Under this respect, the why question has no answer. But, under another point of view, one can say that implication is independently defined by its truth table, and that this truth table explains the equivalence.
Let me try to improve my answer to the second part of your question.
(1) One way to explain why P--> Q is equivalent to ~P v Q is simply to say : " I define a new connective, which I will call implication , P --> Q , as an abbreviation of ~P v Q." In that case, the two expressions are equivalent * by definition* and there is no aswer as to the "why" question : you cannot be wrong when you define the meaning of a term.
Note : in the same way one can say : I define A-B ( substraction ) as an abbreviation for " A + negative B". One cannot possibly be wrong when defining substraction, it's simply a definition. ( This is actually how substraction is defined in mathematics.)
(2) A second way is to say : I have 16 possible binary connectives, one of them has the truth table TT , TF, FT, FF. This connective I will call " material implication" . Once this is done, I notice that it has exactly the same truth table as the expression ~P v Q , and I call P --> Q and ~P v Q " equivalent". This is the semantic way ( since , through the truth tables, you make use of the possible meanings - namely the possible denotations/ truth values - of the sentences.)
(3) A third way is to say. I define P--> Q as ~ (P & ~Q). Then , using rules of transformation , I prove that P --> Q is equivalent to ~P v Q. This is the syntactic way ( since you only and blindly manipulate symbols according to rules, without referring to their possible denotations / truth values).
is equivalent , by definition, to : ~ (P & ~Q)
which in turn is equivalent ( by De Morgan's) : ~P v ~ ~ Q
whic in turn is equivament ( by double negation) to : ~ P v Q.