I think that is not easy to find a good "explanation".
The propositional connectives are a (very simple) mathematical model of natural language, suited for modelling very simple arguments.
Their definition is through truth-table; after you have defined them, you will check how they are "proxing" natural language mechanism.
Someone better (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a "big" approximation : "implies".
I have found useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.
As Kleene says, a lot of controversies aroused around truth-functional definition of "implies".
For me, the traditional locutions : "necessary ... " and "sufficient condition" are a little bit misleading, because they are suggesting a sort of "causal" link between the two statement.
The mathematical model of "if $A$ then $B$" represented by truth-tables does not require any sort of "link" between them.
Assuming now my personal "quasi-conventionalist" reading of the truth-functional connectives, I will try a sort of "reverse engineering" to answer your question.
1) Starting from $A \equiv B$ and agreeing on its "natural" translation as "$A$ if and only if $B$", we have that :
$A \equiv B$ is $A \rightarrow B$ and $B \rightarrow A$.
This is translatable into : "if $A$ then $B$" and "if $B$ then $A$".
But unpacking "if and only if" we have that "$A$ if $B$" and "$A$ only if $B$".
At this point, the "wisdom of the ancients" (see Kleene, pag.63) says that :
"if $A$ then $B$" is "$A$ only if $B$" and that "if $B$ then $A$" is "$A$ if $B$".
The second pair sound more natural to me : into "$A$, if $B$", the "if" is attached to $B$, so it becomes : "if $B$, then $A$".
Then ... les jeux sont fait !
2) And now, what about "sufficient" and "necessary" ?
Let us agree on avoiding the discussion (started in modern times at least from C.I.Lewis, A survey of symbolic logic (1918)) that the truth-fuctional reading of "implies" is not correct, and it is necessary to involve modal concepts in order to correctly explain it.
I think that we must take into account the "isomorphism" between the truth-functional connective "if ... then" and the inference rule of
modus ponens that allows us to infer from the premises $A$ and $A \rightarrow B$, the conclusion $B$.
We must read it as Gottlob Frege did in his Begriffsschrift (1879) :
assuming as true both the premises, the assumption that $A \rightarrow B$ is true, rule-out the row $T-F$ in the truth-table for implies, while the assumption that also $A$ is true rule out two other rows ($F-F$ and $F-T$, respectively). Then, the conclusion that $B$ is true is licensed.
So, assuming the truth-functional definition of "$A$ implies $B$", we have that (the truth of) $A$ is a sufficient condition for (that of) $B$.
See also Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013 - just printed), page 11:
The two sentences if A, then B and B if A seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written $A \rightarrow B$ in the logical notation. Consider the following list :
From A, B follows, A is a sufficient condition for B [...], B is a necessary condition for A, A only if B.
The last two require some thought. The equivalence of $A$ and $B$, $A \leftrightarrow B$ in logical notation, can be read as A if and only if B, also A is a necessary and sufficient condition for B. Sufficiency of a condition as well as the 'if' direction being clear, the remaining direction is the opposite one. So A only if B means $A \rightarrow B$ and so does B is a necessary condition for A.
It sound a bit strange to say that B is a necessary condition for A means $A \rightarrow B$. [...] A necessary condition is instead something that necessary follows, therefore not a condition in the causal sense.