You don't use it like a potato peeler. You don't push it or cut slices with it.
Hold the lemon and the zester roughly vertically, then lay the edge of the zester (the row of little holes) on the lemon. Then pull the zester down across the skin of the lemon. If you apply a little pressure the edge of the zester will scrape the top surface of the peel off the lemon. If you press too hard you will get not just the top surface (the zest) but also the white fluffy layer (the pith). You don't want the pith.
Once you have finished zesting the lemon you won't be down to the fruit part of the lemon, but will have a whole lemon still covered with the white pith, plus a heap of little strips of the zest.
It's like shaving the lemon, rather than peeling it.
(I didn't have a clue either until someone showed me how to do it).