1. What does it mean?
짝남 shortens 짝사랑하는 남자. It refers to the male object of someone's crush, often when the feelings are unspoken or not known to be returned.
Also written or said as jjangnam
2. The nuance
It describes the person being liked, not the speaker's gender. Unlike 남친, it does not mean boyfriend or establish that the two are dating.
3. Examples
“짝남이 좋아하는 책을 나도 읽어 봤어.”
I tried reading a book the guy I have a crush on likes.
“짝남이랑 같은 조가 됐지만 부담 주지 않으려고 평소처럼 얘기했어.”
I ended up in the same group as my crush, but chatted normally so I wouldn't pressure him.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.