I think the easiest way to intuitively appreciate the definition of implication is to not think of statements that are true and false, but of events that "happen" and "don't happen"
So if we think of $A$ and $B$ as events, then we would think of the statement
As saying "Every time $A$ happens, $B$ also happens"
Then false statements correspond to events that never happen, and then it becomes clear why a "false" statement implies anything, for example the truth of the implication
"The moon turns green" $\implies$ "Cats fall from the sky"
Is now much more apparent, since we are saying that every time the moon turns blue, cats start falling from the sky, and this is true in the most basic sense of the word, relating to empirical evidence, because in all our documentation of existence, every time that the moon turned green (which is $0$ times) cats fell from the sky, and we can safely say that every time in the future, when the moon turns green, cats will fall from the sky, and no one will ever be able to prove us wrong, because the moon will never turn green.
This analogy has some drawbacks, mainly that it doesn't really allow you to combine different implications using implication itself, but it gives a feeling for why we chose this definition.
You are also correct that if an implication is true, but the premise is false, then we can't conclude anything from the implication, it is only when you combine a true implication with a true premise that you can conclude the conclusion