It's the opposite of "turn on".
A beautiful woman turns a man on.
An ugly woman turns a man off.
It doesn't necessarily mean "dislike" someone. It simply switches off the nice feelings you had previously.
I just noticed the word order is important. If you turn off somebody, not turn somebody off, then your feelings turn off towards that person. In practice, it all means the same. Watch the word order though.
He's experienced, but his attitude turned me off.
I liked him but then I suddenly turned off him.