An initial Not that clause is short for an extraposed clause. Viz.
- Not that he's a bad boy, ... <== It's not that he's a bad boy, ...
The totally predictable initial It's frequently gets wiped via Conversational Deletion; frequently enough, anyway, to make an idiom.
The intention of the construction is to deny an obvious stereotype judgement by offering evidence of its falsity. Typically the "..." starts with but
- (It's) not that he's a bad boy, but he hangs around with kids that drink.
or continues with a contrasting extraposed clause
- (It's) not that he's a bad boy, it's just that he hangs around with kids that drink.
A frequent variant is Not to say that, which allows one to say something while denying one did.
- Not to say you're wrong, but did you compare the figures with the chart?