You can see in this post the beautiful Henning Makholm's answer :
However, one should note that these [the "usual"] arguments are ultimately not the reason why ⇒ has the truth table it has. The real reason is because that truth table is the definition of ⇒. Expressing p⇒q as "If p, then q" is not a definition of ⇒, but an explanation of how the words "if" and "then" are used by mathematicians, given that one already knows how ⇒ works. The intuitive explanations are supposed to convince you (or not) that it is reasonable to use those two English words to speak about logical implication, not that logical implication ought to work that way in the first place.
I completely agree with him; my personal understanding of this issue is in the answer to this post.
I hope it can help...
For some further insight, I suggest also to see Jan von Plato, Elements of Logical Reasoning (2013), page 97.
With natural deduction for classical logic, we can derive the equivalence between $A \rightarrow B$ and $\lnot (A \land \lnot B)$.
Now, "switching" to the truth conditions for the connectives,
we have that these truth conditions are quite natural for conjunction, disjunction, and negation [...].
For an implication, it is clear that $v(A \rightarrow B) =$ f if $v(A) =$ t and $v(B) =$ f. For the remaining three cases, we can use the classical equivalence of
$A \rightarrow B$ and $¬(A \land ¬B)$ : If $v(¬(A \land ¬B)) =$ t, then $v(A \land ¬B) =$ f by [...] the value of negation. Then either $v(A) =$ f or $v(¬B) =$ f. In the latter case, $v(B) =$ t. Thus, an implication $A \rightarrow B$ has the value t under a given valuation if either $A$ has the value f or $B$ has the value t, otherwise $A \rightarrow B$ has the value f.