For the most accurate answer, you'd better ask Michael Stipe, the singer of REM! ;)
To set something up means that you arranged for it, prepared for it to happen, planned for it, etc. For example, you might set up a meeting or an event.
You can also arrange for something like the punchline of a joke if you tell an entire story only to "set up" the last sentence which reveals the joke.
If you don't prepare properly (e.g. don't study for a test, neglect your responsibilities), we often say "you set yourself up for failure."
In a criminal investigation, a suspect may have been "set up" if a witness lied about them committing the crime. Alternatively, you might set yourself up if you accidentally tell the police too much information which reveals that you are probably guilty.
The REM lyric sounds to me like he revealed too much information, and his secret is now too obvious. Maybe his secret is that he loves the person he's talking about. I'm not sure...