Welcome.
Well 1st of all, there should have been 2X backing installed behind the stone so that there is nailing for the drywall, because drywall shod not be unsupported more than 3".
So, when you have that installed, cut your drywall so that the depressed edge, of the drywall, is next to the stone and the new and old cut edges abutting, I chamfer these to give a greater area for the mud to bond too, but it's not necessary.
I use a light weight setting drywall compound, 20min., for the process, adjacent to the stone, call "flat taping". What you see on, UTUBE, isn't the same.
The Idea is to fill the difference between the two surface so the the horizontal plain of the wall is consistent.
Using a 6" knife you carefully fill the gap, and while you may get some mud on the stone, when it dries it will chip off and a couple of strokes with a damp sponge will take care of the rest. You'll also learn the technique of exerting more or less pressure to the left or right edges of the knife, and how that affects the amount of mud remains.
For the cut edges, there are numerous UTUBE'S for that.