Some a sheet of quare good. stick it to the wall
get a sheet of soft foam, stick it to the wood
stick the darboard to the foam (but fix it so its connected to the wood behind the foam)
That way the darts hitting the dartboard will not cause as much noise as the thudding will go into the foam
clearly you dont wnat the darboard directly touching a wall as that is what is causing the sound to go through to the other room.
getting the right type of foam and making it solid enough so that the dartboard doesnt move is key.