I saw this somewhere a few years ago and had one of those moments my daughter calls a "face-palm". That's where you cover your face with your hand, the way you would if you just discovered that you'd been wearing two different socks all day.
When you open a new box of plastic wrap, start by finding the end of the plastic and pulling off just enough to get it through the open side.
The next part is the really cool thing that I didn't know until, well, adulthood. There are tabs on each end of the box. With the roll in place, push the tabs into the box. They are designed to hold the cardboard tube securely in the box while you pull wrap off the roll.