Moore's Paradox, in the headline, is an effort to remind us how 'the truth' is not simple.
Pontius Pilate long ago asked What is truth? and washed his hands of the whole thing. Professor G E Moore, early last century, proposed his Paradox to get people thinking.
First, the Paradox shows there are facts [It is raining ..] and there are opinions [but I don't believe it ..].
It shows that facts and opinions exist together. And, crucially, that they can contradict each other. A very important reminder for us, considering how often we find them doing so these days.
But it gets more complicated. The sentence, 'It is raining, but I don't believe it' is not nonsense or madness. Not like saying, 'It is raining, but I don't believe it because there is no big tap in the sky.' That would just be silly or insane.
It is not illogical either. It is possible to hold an opinion that does not agree with the facts. The Flat Earth Society still has members, I understand.
Because the sentence is not illogical you cannot defeat it with our normal logic. That is an even more important discovery. There is a contradiction in the sentence, but it is not nonsense nor illogical. How to explain that?
We can point out the person would know it is raining because s/he would sense it: see it, hear it or get wet. But the question then is, are our senses the only way we know things?
What if the person learned it was raining by report? If s/he was told it is raining and said s/he did not believe it, does that alter everything?
Also, if s/he was told it was raining yesterday, but did not believe it, is that a different case again? A bit like History: 'There was a Second World War, but I don't believe it.'
Look at all these problems with the truth. And a professional philosopher would tell me I have hardly started on the subject.