It's a combination of the phrasal verbs "go on" and "be off (to)".
The phrasal verb "go on" means exactly the same thing as "go" in this case. It is only slightly different in that it implies the speaker and listener understand that someone had already planned to go somewhere, and now is the time to go. (Please note that "go on" has idiomatic meanings, as well.)
Go on home, now Johnny. You're mother's waiting for you.
The athletes jumped out of the water and went on to the cycling portion of the triathlon.
The phrasal verb "be off (to)" means to depart for some place, or simply to depart.
I'm off to the store.
Well, it's almost 5 o'clock. I'm off for the day. See you tomorrow.
Where is Julie going? / She's off to the market for some fruit.
So the expression "go on off to work" is simply a combination of these two things. To me, it sounds highly regional and, most likely, rural.