Like half intuitive: $P \implies Q$ is equivalent to $\neg P \lor Q$ because the implication is true when
the premise $P$ is false: Then you can imply everything and whatever
$Q$ is, $P \implies Q$ is true. In other words: A false statement can
imply a false statement and a true statement.
the conclusion $Q$ is true: Both a true and a false statement can imply a true statement. So when the conclusion is true, the implication is true - whatever the premise $P$ is.
So the negation has to be $\neg(\neg P \lor Q)$ and that is $P \land \neg Q$.
Your real world example:
today is sunny $\implies$ you go to the sea
If this implication is true, what are the cases which can occur?
It is sunny and you go to the sea. Yeah. That's the trivial case.
It isn't sunny and you don't go to the sea. That's possible. We only made a statement about the sunny situation.
It is sunny and you don't go to the sea. NO! We said, if it is sunny (and that is the case), you go to the sea.
It isn't sunny and you go to the sea. That is the thing: That's possible. We said you will you to the sea if it is sunny. But we didn't say something about what you do when it is not sunny. If it's sunny, you go, if it is not - nobody knows.
If it is sunny, I know that you went to the sea. But when you are at the sea, I don't know if it is sunny (that means I cannot conclude that it is sunny when I see you at the sea).
What is now the negation of the implication? That is exactly the third statement. We said, always when the sun shines, you go to the sea. So the correct negation is the case that the sun shines but the implication isn't fulfilled - you don't go to the sea.
Let $P$ be it is sunny and let $Q$ be you go to the sea, then
$$ \neg(P \implies Q) \iff P \land \neg Q $$
as we saw.
That is not the same as $P \implies \neg Q$ which means always when it is sunny, you do not go to the sea. We only considered the negation of the one implication (not always when it is sunny, you go to the sea) and didn't make a statement about all non-sunny days (always when it is sunny, you don't go to the sea is something different!).
The negation sais: There is one situation where it is sunny (that's $P$) but you do not go to the sea (that's $\neg Q$). So it is not the case that $P \implies Q$.