What do you think people are really saying when they say they are special and different or don't fit in?

They might genuinely feel that way. I think that the first statement is slightly different to the others. A feeling of being special is born of confidence or arrogance. It could be narcissism or they may simply feel that they have a talent and a higher purpose. Feeling different or like you don't fit is indicative of lacking a sense of belonging and could be born of social anxiety or mistreatment. The reality is that none of us are really that different or unique in the broader sense, but some people really are special and that's why we have geniuses, Olympians, professional sports stars, etc.