Hi,
it's probably because "edit button" is connected with specific category and you need to login first.
It's has misleading name, because in Menu Manager -> Menu Item Type : "K2 » Item edit form" - is not for editing articles but for creating a new one. Because after clicking on this menu item you'll get new window with blank article form.
If you want to edit articles use module from K2 and/or
second solution login first and go where you have K2 article open.
k2-edit.png
what's that???
Ask creators/developers of K2 why they build in this way. Nobody said that creating by K2 will be easy