On and Off when used in those kinds of situations are kind of like prepositions, they indicate a direction related to the verb. As Sarapva said, "I'm off!" indicates the act of exiting or leaving because 'off' as a preposition usually indicates a departure of some kind. To 'get off a train', 'get off a horse', even something action-themed like "I managed to get a shot off but I didn't hit him" referring to the firing of a gun.. the shot got off, as in, the bullet was issued (away from the shooter.)
There is no linear relationship in some cases, however. While you can get on and off a train or bicycle, you cannot refer to the same action situation with the gun, because bullets never enter a gun when fired they only exit. "Go on"... the on there I *think* is an abbreviated form of 'onward', as in continuing the present course or direction. During an explanation or story, if the speaker pauses, the listener might say "Go on", or if physically walking, you could also tell someone who's stopped to "go on, keep moving".
In my mind, the idea behind 'go on home' follows this trend. However, there is no real opposite to 'onward' that translates directly into off.. i.e. there is no 'offward'. Therefore 'go off home' doesn't mean the same thing OR the exact opposite thing. Most commonly, to 'go off' means something completely different.. i.e. to detonate, explode, finish counting down, like a bomb or alarm clock.