The phrase has part of its origins in the police interrogation tactic of threatening criminals who will not cooperate. Often the only reason a perpetrator doesn't snitch is because he fears it will get to the street and put him in danger. Ironically police officers threaten to publicize or have correctional officers publicize that a perpetrator's "jacket" says they are an informant to get them to inform. Someone with a "snitch jacket" is often just a willing informant obviously.
DillyDuckets: Nurga had his snitch jacket on to the fullest.
BillyBadass: His days of ridin candy slab are over.
Con2: "Yea his homies gave him a snitch jacket for rattin them out"