It's really "I'd just as soon...", meaning "I would just as soon..." It is usually followed by some possible activity that clearly no one would think of doing. An example might be "Watch a football match? I'd just as soon rip my eyeballs out!"
In other words, it means that - whatever the original suggested activity is - this person has no interest whatsoever in doing it. So much so, in fact, that he'd be as likely to do the other thing.