Conditional statements?

It's important to realize that $p \implies q$ is not any kind of causative relationship - it's not saying $p$ causes $q$, or anything like that. It's saying "if $p$ happens to be true, then $q$ also happens to be true, possibly by pure coincidence".

Likewise, "necessary condition" and "sufficient condition" don't say anything about causation. $q$ is a necessary condition for $p$ exactly if $p$ can only be true if $q$ is; $p$ is a sufficient condition for $q$ exactly if whenever $p$ is true, so is $q$ - possibly coincidentally.

So suppose we know that the sentence $p \implies q$ is true. Then we know that if $p$ happens to be true, so does $q$; that's what the $\implies$ symbol means. So $p$ can't be true if $q$ isn't - if it were the case that $p$ is true with $q$ false, our sentence $p \implies q$ would be false. But that's the definition of $q$ being a necessary condition for $p$! Likewise, $q$ has to be true whenever $p$ is, which means $p$ is a sufficient condition for $q$.

Notice that it doesn't work the other way around - $p \implies q$ doesn't mean $p$ is a necessary condition for $q$, because $q$ might be implied by other, unrelated things as well.

We remind that if $p \implies q$, then: - $q \implies - p$

These are other equivalent ways to say $p \implies q$ in English. Personally I think "$p$ implies $q$" is the clearest, but I suppose it is important to understand the other phrases, since people use them often. Here is the intended interpretation.

"$p$ is a sufficient condition for $q$": if you want to know if $q$ is true, then it is enough (sufficient) to know that $p$ is true. However, there may be other ways for $q$ to be true, so in this case you cannot say $p$ is a necessary condition for $q$. [As an example, if $p$ is "my favorite number is divisible by $4$" and if $q$ is "my favorite number is even," then knowing $p$ is true is enough (sufficient) to conclude that $q$ is true, but there are other ways for $q$ to be true without $p$ being true, e.g. the case where my favorite number is $2$.]

"$q$ is a necessary condition for $p$": simply, if $p$ is true, then $q$ must (necessarily) also be true. It is necessary in the sense that if $q$ were not true, then it would be impossible for $p$ to be true.