The term was taken from the movie starring Steve Martin titled The Jerk. Throughout the film Steve Martins Character is stalked by a bumbling fat ass grease ball of a raving, militant, lunatic, sniper, serial killer who chose his victims this way. Upon completing the process the killer exclaimed "I got you now you Random Bastard, son of a bitch." and referred to his targets in this way mumbling under his breath and stalking them throughout the movie.
Generally it infers an underlying contemptuous or punitive tone.
An example of its use in a driving situation, shouted, uttered or mumbled by someone who politely drives down the road all the while cussing at everyone who crosses their path. (and closet smoking)
Hang up and drive!, get the hell over! DIE YOU RANDOM BASTARD!