Ah, English language, I’ll never not love you.
Negatives are powerful things. “No” is one of the first words we learn to understand as babies (apparently) and is one of the things which protects our identity and independence as we get older. Yes is often easier to say, it makes you a follower; No makes you stand out.
Negatives can have positive synonyms, but while the information they convey can be identical, the weight and feel of the message changes. Sometimes dramatically. Compare: “I will see him next year” with “I won’t see him until next year” or “I’m leaving him behind” with “I’m not taking him with me.”
Double negatives are often frowned upon and frequently misused, but they are an enriching part of English idiom too. To understand the technical meaning of a double negative, you can just take them both out, but the richness comes because it’s so more complicated than that!
“I ain’t never eating brocolli again,” technically ought to mean “I am always going to eat broccoli” but of course it doesn’t. It’s an emphatic way of saying “I’m never eating broccoli again” with a hint at the geographical or class origins of the speaker.
Conversely, there’s a line in “The Wizard and I” from Wicked which goes “no father is not proud of you”. This time, it’s technically correct. The double negative is used to mean “Your father is proud of you”, but the use of the double negative gives a clear image of the emotional state and history of the character. “Your father is proud of you” is pretty neutral, whereas the line as it is used is loaded with emotion – the singer’s father isn’t currently proud of her, she resents this and dreams of a world in which things are different.