Like others said, it's a colloquial way of talking. Very informal in English.
Listen to the tone in which someone says something like that. If they're speaking very definitively, hear the "no" sound in their tone. They're not going to talk.
This is rare, but if someone says, very carefully, carefully emphasizing certain words, then they might be saying, "I'm saying something, but you need to work carefully to get my meaning." I'm not saying nothing, my words are not nonsense, I'm saying something, but I can't say it directly.