In your example the phrase "to go off the grid" refers to the idea of hiding from all observation or tracking. For example, if someone commits a crime, they might attempt to "go off the grid" by avoiding the use of their credit cards, cell phone, and anything else linked to their identity. This would make it difficult for the authorities to find them. There are a few other common phrases with a similar meaning. If we imagine the same situation of a criminal wanting to hide they might: "Lay low"/"Lie low" "Keep a low profile" "Go into hiding" All of these phrases imply a similar idea of hiding or avoiding attention, and all are very common in certain genres of films etc.

I would say all these answers are great. If you want an example of someone who has gone off the grid, but not for criminal reasons, Henry David Thoreau is probably the most famous example. He lived in a cabin in the woods for many years with very little to no contact with anyone else and wrote Walden while doing so.

Going off the grid is colloquial and it could be taken a few ways. One of them is to go off the utilities (power, water, telecommunications) grid, that is to say, the household lives not connected to wires and pipes, perhaps in the woods, and the household is self sustaining. Perhaps another way to look at it is that you leave telecommunications behind and take a vacation. "I went to Alaska and hiked off the grid in the isolated backwoods." You got me curious. This web link describes off the grid from "Urban Dictionary." I am not familiar with the site, but what it says makes sense to me. If it helps, I am a a native mid-western American. Best wishes for your studies. https://www.urbandictionary.com/define.php?term=off%20the%20grid

Ed and Sean answered your question very well. "Go off the grid" is part of a larger structure. "Go adj" or "go prepositional phrase" means that you change from NOT being something to BEING that thing. It is a spoken version of "become". "Bob Used to be very well-behaved, but after his parents died, he went wild."=he became wild=he changed from not being wild to being wild. "When I told Sam that I had lost his money, he went out of his mind"=he went crazy=he became crazy=he changed from not being crazy to being crazy.