The mystery machine is turning items right-side out after they were sewn together while inside out. It's much cleaner and easier to use a fine-tipped tool like that than shoving your grubby hand inside the item and flailing around trying to grab the corner to pull it through.
7:20 they have stitched up a piece of something then they are turning the points of the garment or stuffed toy or whatever, so that the excess material will be on the inside. Basically they are turning the The sewn object right side out.
You asked what the person was doing with that white fabric and those two long needles. Looks like they’re turning inside out and inner pocket of a garment, to get a nice sharp inner crease.