At first you could say there is something sweet about this song. It's not that he's jealous, he just wants to protect her, keep her away from people with diseases and all that. He doesn't want her back, he knows that they don't work, but he still wants to kill whatever motherfucker is trying to bed her currently.
But as the song plays on (and any guy listening to this should get it way before he reveals it in the song) you learn he's lying. He's not protecting her, he's trying to possess her still, insisting he doesn't want her but all the while trying to guide her back into his good arms.
A guy on Songmeanings points out that it sounds like, from the lyrics, that he broke up with her ("I decided this decision some six months ago") and can't handle watching her move on. I didn't get that, though it makes perfect sense, so I am sharing it.
I dumped a girl who I still wanted once and it totally ruined the relationship. Also, everyone in the whole world can say "I can relate to wanting to kill someone else's partner," so I don't really need to say that.
P.S. This is a pretty song.