I cant remember if it was explained, but I think it would cause conflict to the balance of the earth. MC just basically unintentionally cross dimensions. If there is a crack in the dimension, and those who are in the magical world would be aware of it. It would be dangerous if magical world criminals would cross dimensions or something. Especially because the world where MC is from only has humans, and dont have magical abilities.
From my memory it wasn't the world traveller part that was the problem because the convenience store is travelling to other spaces all the time. The problem was that he shouldn't have been able to enter because the manager didn't go to his world - his world is not part of their dimension. So I believe the manager wanted to keep it secret so he didn't get turned into an experimental lab rat of sorts. They have humans in the dimension but he isn't a human from that dimension. It's also why he doesn't go exploring the world beyond the convenience store and getting his certification. This is all from memory though so I may be wrong or have jumbled stories...
It might seem obvious, but why do MC need to keep his origin a secret? This is a magical world, right? Is a world traveller that much of a stretch? Has it been explained and I forgot?