How to properly use a butter stick's wrapper?
I would suggest one of:

1) Cut the butter with a sharp knife (if cold enough) and throw away paper on portion you use. Then take a small square of aluminum foil to cover the end (or cellophane).

2) Do the above but don't worry about the foil. Butter will keep fine in the fridge uncovered.

3) If you're not keeping your butter in the fridge then I'd suggest just getting a porcelain butter keeper for it and use what you need. I've tried the ones that you put into water and I ended up with a disgusting mold factory. 

4) Use a butter keeper in the fridge. I clean mine with hot water and then dry with a towel and put a new stick of butter on it while it's still warm. The bottom of the butter melts slightly and then sticks to the porcelain so it's easier to cut off a piece when cold.

5) I've seen Gordon Ramsay use the aluminum butter wrapper to wrap a piece of veal that he then puts into the oven. Who knew?